Essential Self-Adjointness of Even-Order, Strongly Singular, Homogeneous Half-Line Differential Operators

Abstract

We consider essential self-adjointness on the space $C_0^{\infty }((0,\infty ))$ of even-order, strongly singular, homogeneous differential operators associated with differential expressions of the type

$$\begin{aligned} \tau _{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x > 0, \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$

in $L^2((0,\infty );dx)$. While the special case $n=1$ is classical and it is well known that $\tau _2(c)\big |_{C_0^{\infty }((0,\infty ))}$ is essentially self-adjoint if and only if $c \ge 3/4$, the case $n \in {{\mathbb {N}}}$, $n \ge 2$, is far from obvious. In particular, it is not at all clear from the outset that

$$\begin{aligned} \begin{aligned}&\textit{there exists }c_n \in {{\mathbb {R}}}, n \in {{\mathbb {N}}}\textit{, such that} \\&\quad \tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))} \, \textit{ is essentially self-adjoint}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (*)\\ {}&\quad \textit{ if and only if } c \ge c_n. \end{aligned} \end{aligned}$$

As one of the principal results of this paper we indeed establish the existence of $c_n$, satisfying $c_n \ge (4n-1)!!\big /2^{2n}$, such that property (*) holds. In sharp contrast to the analogous lower semiboundedness question,

$$\begin{aligned} \textit{for which values of }c\textit{ is }\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}{} \textit{ bounded from below?}, \end{aligned}$$

which permits the sharp (and explicit) answer $c \ge [(2n -1)!!]^{2}\big /2^{2n}$, $n \in {{\mathbb {N}}}$, the answer for (*) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly,

$$\begin{aligned} c_{1}&= 3/4, \quad c_{2 }= 45, \quad c_{3 } = 2240 \big (214+7 \sqrt{1009}\,\big )\big /27, \end{aligned}$$

and remark that $c_n$ is the root of a polynomial of degree $n-1$. We demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as radicals over ${{\mathbb {Q}}}$ (and conjecture this is in fact true for general $n \ge 6$).

1 Introduction

Consider the 2nth-order differential expression

$$\begin{aligned} \tau _{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x \in (0,\infty ), \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$

(1.1)

and introduce the underlying preminimal and symmetric $L^2((0,\infty ); dx)$-realization

$$\begin{aligned} \tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))} \end{aligned}$$

(1.2)

and its closure, the associated minimal operator $T_{2n, min}(c)$ in $L^2((0,\infty ); dx)$,

$$\begin{aligned} T_{2n, min}(c) = \overline{\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}}. \end{aligned}$$

(1.3)

The principal question to be posed and answered in this paper is the following:

$$\begin{aligned}&\textit{ For which values of }c \in {{\mathbb {R}}}\textit{ is }T_{2n, min}(c)\textit{ self-adjoint }(\textit{equivalently,} \nonumber \\&\quad \textit{ for which values of }c \in {{\mathbb {R}}}\textit{ is }\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}{} \textit{ essentially self-adjoint}\,) \nonumber \\&\quad \textit{ in }L^2((0,\infty ); dx)? \end{aligned}$$

(1.4)

For the notion of (essentially) self-adjoint Hilbert space operators see, for instance, [27, Sect. V.3], [39, Sect. VIII.2], [45, Sect. 3.2], and [52, Sects. 4.4, 5.3].

In the special case $n=1$ it is well known that the precise answer is (see, e.g., [46]),

$$\begin{aligned} (1.4)\textit{ holds for }n=1\textit{ if and only if } \, c \ge c_1 = 3/4. \end{aligned}$$

(1.5)

A priori it is not clear at all that this extends to $n \in {{\mathbb {N}}}$, $n \ge 2$, that is, it is not obvious from the outset that

$$\begin{aligned} \begin{aligned}&\textit{ there exists }c_n \in {{\mathbb {R}}}, n \in {{\mathbb {N}}}\textit{, such that} \\&\quad \tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))} \, \textit{ is essentially self-adjoint if and only if }c \ge c_n. \end{aligned} \end{aligned}$$

(1.6)

Our principal new results, Theorem 4.5 and Corollary 4.7 assert that (1.6) indeed holds for some $c _n \in {{\mathbb {R}}}$ satisfying

$$\begin{aligned} c_n \ge (4n-1)!!\big /2^{2n}, \quad n \in {{\mathbb {N}}}. \end{aligned}$$

(1.7)

The proof of the existence of $c_n$ in (1.6) (satisfying (1.7)) is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. Explicitly, one obtains

$$\begin{aligned} c_{1}&= 3/4, \quad c_{2 }= 45, \quad c_{3 } = 2240 \big (214+7 \sqrt{1009}\,\big )\big /27, \nonumber \\ c_{4 }&= 2835 \Bigg ( 13711+\frac{190309441}{\root 3 \of {2625188010911+1805760 \sqrt{-292868607}}} \nonumber \\&\quad + \root 3 \of {2625188010911+1805760 \sqrt{-292868607}}\ \Bigg ) \end{aligned}$$

(1.8)

and we note that in this context that $c_n$ is the root of a polynomial of degree $n-1$. In addition, we demonstrate that for $n=6,7$, $c_n$ are algebraic numbers not expressible as radicals over ${{\mathbb {Q}}}$; we conjecture that this actually continues to hold for general $n \ge 6$.

Before explaining some of the strategy behind the proof of the existence of $c_n$, and for the purpose of comparison and exhibition of a sharp contrast to the essential self-adjointness problem (1.6), we briefly record the precise borderline of semiboundedness of the minimal operator $T_{2n, min}(c)$, which permits a remarkably simple and explicit solution as follows:

$$\begin{aligned}&T_{2n, min}(c)\textit{ is bounded from below, and then actually, }T_{2n, min}(c) \ge 0, n \in {{\mathbb {N}}}, \nonumber \\&\quad \textit{ if and only if } \, c \ge - \frac{[(2n -1)!!]^{2}}{2^{2n}}. \end{aligned}$$

(1.9)

This is a consequence of the sequence of sharp Birman–Hardy–Rellich inequalities, see Birman [5, p. 46] (see also Glazman [17, p. 83–84])

$$\begin{aligned} \begin{aligned} \int _0^{\infty } \textrm{d}x \, \big | f^{(n )}(x)\big |^{2}&\ge \frac{[(2n -1)!!]^{2}}{2^{2n}} \int _0^{\infty } \textrm{d}x \, x^{-2n} |f(x)|^2, \\ f&\in C_{0}^{n }((0,\infty )), \; n \in {{\mathbb {N}}}. \end{aligned} \end{aligned}$$

(1.10)

For more details on (1.10) see [16] and the extensive literature cited therein.

Returning to (1.6), our subject at hand, we recall that $\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}$ is essentially self-adjoint in $L^2((0,\infty ); dx)$ if and only if $\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}$ is in the limit point case at $x=0$ and $x = \infty$. However, since for all $c \in {{\mathbb {R}}}$, $c x^{-2n}$ is bounded on $(\varepsilon ,\infty )$ for all $\varepsilon > 0$, $\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}$ is automatically in the limit point case at $x = \infty$ and hence it suffices to exclusively focus on whether or not $\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}$ is in the limit point case at $x=0$.

In this context one observes that $\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}$ is said to be in the limit point case at an interval endpoint $a \in \{0, \infty \}$ if precisely n solutions of

$$\begin{aligned} \tau _{2n}(c) y(\mu ,\,\cdot \,;c) = \mu y(\mu ,\,\cdot \,;c), \quad \mu \in {{\mathbb {C}}}\backslash {{\mathbb {R}}} \end{aligned}$$

(1.11)

(i.e., precisely half of the solutions) lie in $L^2(I_a; dx)$, where $I_a$ is an interval of the type $I_0 = (0,d)$ if $a=0$, and $I_{\infty } = (d,\infty )$ if $a=\infty$, for some fixed $d \in (0,\infty )$.

To decide the limit point property of $\tau _{2n}(c)\big |_{C_0^{\infty }((0,\infty ))}$ at $x=0$, one next argues that it is possible to choose $\mu = 0$ in (1.11), restricting x to the interval $I_0=(0,d)$, which then leads to a special Euler-type equation which generically has solutions of power-type

$$\begin{aligned} y_j(0,x;c) = C_j x^{\alpha _j(c)}, \quad 1 \le j \le 2n, \end{aligned}$$

(1.12)

with $\alpha _j(c)$, $1 \le j \le 2n$, being the solutions of the underlying discriminant or indicial equation,

$$\begin{aligned} D_{2n}(z;c) = \prod _{j=1}^{2n} [z - (j-1)] + (-1)^n c = 0, \quad z \in {{\mathbb {C}}}. \end{aligned}$$

(1.13)

In exceptional cases, where some of the $\alpha _k(c)$ coincide, (1.12) is replaced by

$$\begin{aligned} y_k(0,x;c) = C_k x^{\alpha _k(c)} P({\text {ln}}(x)), \end{aligned}$$

(1.14)

where $P(\,\cdot \,)$ is a polynomial of degree at most $2n-1$. Since we are interested in whether or not $y_j(0,x;c) \in L^2((0,d);dx)$ for some $d \in (0,\infty )$, the presence of logarithmic terms is irrelevant and the deciding $L^2$-criterion for solutions of $\tau _{2n}(c) y(\mu ,\,\cdot \,;c) = 0$ simply becomes

$$\begin{aligned} \begin{aligned}&{\text {Re}}(\alpha _j(c)) > - 1/2, \, \text { for }L^2\text {-membership}, \\&\quad \text {respectively, } \, {\text {Re}}(\alpha _j(c)) \le - 1/2, \, \text { for non-}L^2\text {-membership.} \end{aligned} \end{aligned}$$

(1.15)

In conclusion, to settle the essential self-adjointness problem (1.6) one needs to establish the existence of $c_n \in {{\mathbb {R}}}$ such that precisely n roots $\alpha _j(c)$ of $D_{2n}(\,\cdot \,;c)=0$ satisfy ${\text {Re}}(\alpha _j(c)) \le - 1/2$ for $c \ge c_n$. (Equivalently, precisely n roots $\alpha _k(c)$ of $D_{2n}(\,\cdot \,;c)=0$ satisfy ${\text {Re}}(\alpha _k(c)) > - 1/2$ for $c \ge c_n$.)

Turning briefly to the content of each section, we note that Sect. 2 introduces minimal and maximal operators associated with general differential expressions $\tau _{2n}$ of order 2n, $n \in {{\mathbb {N}}}$, in $L^2((0,\infty ); dx)$ and reviews the underlying facts on deficiency indices of the minimal operator $T_{2n,min}$, including Kodaira’s decomposition principle. Section 3 discusses perturbed Euler differential systems and investigates the underlying deficiency indices for the minimal operator associated with $\tau _{2n}(c)$ in (1.1). In addition, some of the basic theory of first-order systems in the complex domain going back to Fuchs, Frobenius, and Sauvage, in versions championed by Hille and Kneser, is summarized. Moreover, the special examples $\tau _2(c)$ and $\tau _4(c)$ are treated explicitly. Properties of the (real part of the) roots $\alpha _j(c)$ of $D_{2n}(\,\cdot \,;c)=0$ are the center piece of our principal Sect. 4, culminating in Theorem 4.5 and Corollary 4.7 which settle the essential self-adjointness problem (1.6). The techniques involved are related to the Grace–Heawood theorem [38, p. 126], the Routh–Hurwitz criterion, and Orlando’s formula [13, § XV.7]. Appendix A shows with the help of Galois theory that $c_6$ is an algebraic number that cannot be expressed as radicals over ${{\mathbb {Q}}}$; we conjecture this actually remains the case for all $c_n$, $n \in {{\mathbb {N}}}$, $n \ge 6$.

Finally, some remarks on the notation employed: We denote by ${{\mathbb {C}}}^{M \times N}$, $M, N \in {{\mathbb {N}}}$, the linear space of $M \times N$ matrices with complex-valued entries. $I_N$ represents the identity matrix in ${{\mathbb {C}}}^N$. The spectrum of a matrix (or closed operator in a Hilbert space) T is denoted by $\sigma (T)$. The abbreviation ${{\mathbb {N}}}_0= {{\mathbb {N}}}\cup \{0\}$ is used.

2 The Deficiency Indices of $T_{2n, min}(c)$

In this section we briefly recall the notions of deficiency indices and limit point, respectively, limit circle cases associated with maximally defined differential operators, generated by formally symmetric differential expressions $\tau _{2n}$ on intervals $(a,b) \subseteq {{\mathbb {R}}}$, of even order 2n, $n \in {{\mathbb {N}}}$, and then specialize the results to the particular case $\tau _{2n}(c)$ at hand. We will primarily follow [7, Sects. XIII.2, XIII.6], [34, Sects. 17.4, 17.5], [53, Sects. 3, 4] and also refer to [2, § 126], [23, 24, 29, 30, 51, Chs. 2–4] for relevant background material.

Assuming $(a,b) \subseteq {{\mathbb {R}}}$ we suppose that

$$\begin{aligned}&p_m, r \text { are (Lebesgue) measurable and real-valued a.e. on }(a,b), \quad 0 \le m \le n, \nonumber \\&p_n> 0, \; r > 0 \text { (Lebesgue) a.e. on }(a,b), \nonumber \\&(1/p_n), \; p_m \in L^1_{loc}((a,b); dx), \quad 0 \le m \le n-1, \end{aligned}$$

(2.1)

and introduce the quasi-derivatives

$$\begin{aligned} u^{[0]}&= u, \; u^{[m]} = u^{(m)}, \quad 0 \le m \le n-1, \nonumber \\ u^{[n]}&= p_n \big (u^{{n-1}}\big )', \nonumber \\ u^{[n+1]}&= - \big (u^{{n}}\big )' + p_{n-1} u^{{n-1}}, \nonumber \\ u^{[n+j]}&= - \big (u^{{n+j-1}}\big )' + p_{n-j} u^{{n-j}}, \quad 2 \le j \le n -1, \nonumber \\ u^{[2n]}&= - \big (u^{{2n - 1}}\big )' + p_0 u = r (\tau _{2n} u). \end{aligned}$$

(2.2)

Here the formally symmetric differential expression $\tau _{2n}$ of order 2n is given by

$$\begin{aligned} (\tau _{2n} u)(x) = \sum _{m=0}^n (-1)^m \big (p_m(x) y^{(m)}(x)\big )^{(m)}, \quad x \in (a,b). \end{aligned}$$

(2.3)

Given (2.1)–(2.3), the maximal $L^2((a,b); rdx)$-realization (in short, the maximal operator), $T_{2n,max}$, associated with $\tau _{2n}$ is then defined by

$$\begin{aligned} T_{2n, max} f=&\tau _{2n} f, \nonumber \\ f \in {\text{ dom }}(T_{2n, max})=&\big \{g \in L^2((a,b);rdx) \, \big | \, g^{[\ell ]} \in AC_{loc}((a,b)), \, 0 \le \ell \le 2n - 1; \nonumber \\ {}&\qquad \qquad \tau _{2n} g \in L^2((a,b);rdx)\big \}. \end{aligned}$$

(2.4)

Introducing the preminimal operator

$$\begin{aligned} \begin{aligned} \overset{{\textbf {.}}}{T}_{2n, min} f&= \tau _{2n} f, \\ f \in {\text {dom}}\big (T_{2n, min}\big )&= \{g \in {\text {dom}}(T_{2n, max}) \, | \, {\text {supp}}\,(g) \, \text {compact}\} \end{aligned} \end{aligned}$$

(2.5)

in $L^2((a,b); rdx)$, one can show that $\overset{{\textbf {.}}}{T}_{2n, min}$ is densely defined, symmetric, and closable. Hence, defining the minimal operator in $L^2((a,b); rdx)$ associated with $\tau _{2n}$ as the closure of $\overset{{\textbf {.}}}{T}_{2n, min}$,

$$\begin{aligned} T_{2n, min} = \overline{\overset{{\textbf {.}}}{T}_{2n, min}}, \end{aligned}$$

(2.6)

one can prove the well-known fact

$$\begin{aligned} T_{2n, min}^* = T_{2n, max}, \quad T_{2n, max}^* = T_{2n, min}, \end{aligned}$$

(2.7)

and thus $T_{2n, max}$ is closed. Moreover, if

$$\begin{aligned} p_m \in C^m((a,b)), \quad 0 \le m \le n, \end{aligned}$$

(2.8)

one can introduce

$$\begin{aligned} {\overset{{\textbf {..}}}{T}}_{2n, min} = \tau _{2n}\big |_{C_0^{\infty }((a,b))}, \end{aligned}$$

(2.9)

and then also obtains

$$\begin{aligned} \overline{{\overset{{\textbf {..}}}{T}}_{2n, min}} = \overline{\overset{{\textbf {.}}}{T}_{2n, min}} = T_{2n, min}. \end{aligned}$$

(2.10)

Introducing the Lagrange bracket

$$\begin{aligned}{}[u,v]_x = \sum _{j=1}^n \big [u^{[j-1]}(x) v^{[2n-j]}(x) - u^{[2n-j]}(x) v^{[j-1]}(x)\big ], \quad x \in (a,b),\qquad \quad \end{aligned}$$

(2.11)

one infers for $(d,e) \subset (a,b)$ Lagrange’s identity via integrations by parts

$$\begin{aligned}{} & {} \int _d^e r(x) dx \, \big \{\overline{(\tau _{2n} u)(x)} v(x) - \overline{u(x)} (\tau _{2n} v)(x)\big \} \nonumber \\{} & {} \quad = [\overline{u},v]_e - [\overline{u},v]_d = [u,v]_x\big |_{x=d}^e. \end{aligned}$$

(2.12)

Moreover, if $u(\overline{\mu }, \,\cdot \,)$ and $v(\mu , \,\cdot \,)$ are solutions of

$$\begin{aligned} (\tau _{2n} u(\overline{\mu }, \,\cdot \,))(x)= & {} \overline{\mu }u(\overline{\mu },x), \quad \nonumber \\ (\tau _{2n} v(\mu , \,\cdot \,))(x)= & {} \mu v(\mu ,x), \quad \mu \in {{\mathbb {C}}}, \; x \in (a,b), \end{aligned}$$

(2.13)

then

$$\begin{aligned} \frac{d}{dx} [\overline{u(\overline{\mu }, \,\cdot \,)}, v(\mu ,\,\cdot \,)]_x = 0, \quad x \in (a,b). \end{aligned}$$

(2.14)

Finally, we also recall the known fact,

$$\begin{aligned} \begin{aligned} {\text{ dom }}(T_{2n, min})=&\{g \in {\text{ dom }}(T_{2n,max}) \, | \,\text{ for } \text{ all } h \in {\text{ dom }}(T_{2n,max}): \\ {}&\qquad [h,g]_a= 0 = [h,g]_b \, \}. \end{aligned} \end{aligned}$$

(2.15)

In the following, the number of $L^2((a,b); rdx)$-solutions $u(\mu _{\pm }, \,\cdot \,)$ of

$$\begin{aligned} \tau _{2n} u(\mu _{\pm }, \,\cdot \,) = \mu _{\pm } u(\mu _{\pm }, \,\cdot \,), \, \text { with }\pm {\text {Im}}(\mu _{\pm }) > 0, \end{aligned}$$

(2.16)

is denoted by $n_{\pm } (T_{2n, min})$ and called the deficiency indices of $T_{2n, min}$. This notion is well defined as $n_{\pm } (T_{2n, min})$ is known to be constant throughout the open complex upper and lower half-plane. As a result, one typically chooses $\mu _{\pm } = \pm i$. Since the coefficients of $\tau _{2n}$ are real-valued, one obtains by a result of von Neumann [49] that

$$\begin{aligned} 0 \le n_+ (T_{2n, min}) = n_- (T_{2n, min}) \le 2n. \end{aligned}$$

(2.17)

Finally, given $d \in (a,b)$, and denoting by $T_{2n, min (max),(a,d)}$ and $T_{2n,min (max),(d,b)}$ the corresponding minimal or maximal operator with the interval (a, b) replaced by (a, d) and (d, b), respectively, where d is now a regular endpoint for $\tau _{2n}\big |_{(a,d)}$ and $\tau _{2n}\big |_{(d,b)}$, one has (cf. [2, p. 483–484])

$$\begin{aligned} n_+ (T_{2n, min, (a,d)})&= n_- (T_{2n, min, (a,d)}), \quad n_+ (T_{2n, min, (d,b)}) = n_- (T_{2n, min, (d,b)}), \nonumber \\ n&\le n_{\pm } (T_{2n, min, (a,d)}) \le 2n, \quad n \le n_{\pm } (T_{2n, min, (d,b)}) \le 2n, \end{aligned}$$

(2.18)

and the Kodaira decomposition principle (see, e.g., [7, Corollary XIII.2.26], [34, p. 72])

$$\begin{aligned} n_{\pm } (T_{2n, min}) = n_{\pm } (T_{2n, min, (a,d)}) + n_{\pm } (T_{2n, min, (d,b)}) - 2n \end{aligned}$$

(2.19)

holds.

Remark 2.1

Given the fact that $d \in (a,b)$ is a regular endpoint for $\tau _{2n}|_{(a,d)}$ and $\tau _{2n}|_{(d,b)}$, the particular (and extreme) case where

$$\begin{aligned} n_{\pm } (T_{2n, min, (a,d)}) = n \, \text { (resp., }n_{\pm } (T_{2n, min, (d,b)}) = n) \end{aligned}$$

(2.20)

is the precise analog of Weyl’s limit point case at $x=a$ (resp., $x=b$) in the classical second-order case $n=1$, that is, for $\tau _2|_{(a,d)}$ (resp., $\tau _2|_{(d,b)}$). Hence, we will apply this limit point terminology also in the 2nth-order context in the following. In particular, if

$$\begin{aligned} n_{\pm } (T_{2n, min, (a,d)}) = n = n_{\pm } (T_{2n, min, (d,b)}), \end{aligned}$$

(2.21)

then $\tau _{2n}|_{(a,b)}$ is in the limit point case at a and b and (2.19) yields accordingly that

$$\begin{aligned} n_{\pm } (T_{2n, min}) = 0 \end{aligned}$$

(2.22)

in this case. Thus, (2.21), and hence (2.22), is equivalent to

$$\begin{aligned} T_{2n, min} = T_{2n, max} \, \text { is self-adjoint in }L^2((a,b); rdx), \end{aligned}$$

(2.23)

which in turn is equivalent to

$$\begin{aligned} \overset{{\textbf {.}}}{T}_{2n, min} \, \text { is essentially self-adjoint in }L^2((a,b); rdx). \end{aligned}$$

(2.24)

If in addition hypothesis (2.8) holds, then each of (2.21)–(2.24) is also equivalent to

$$\begin{aligned} {\overset{{\textbf {..}}}{T}}_{2n, min} \, \text { is essentially self-adjoint in }L^2((a,b); rdx). \end{aligned}$$

(2.25)

All other cases, where $1 \le n_{\pm } (T_{2n, min}) \le 2n$, describe various degrees of limit circle cases of $\tau _{2n}$, with $n_{\pm } (T_{2n, min}) = 2n$ representing the extreme case. $\diamond$

In the bulk of this paper we are particularly interested in the special case where

$$\begin{aligned} p_n(x){} & {} =1, \quad p_m(x) = 0, \;1 \le m \le n-1, \quad p_0(x) = c x^{- 2n}, \nonumber \\{} & {} \quad r(x) = 1, \quad x \in (0,\infty ), \end{aligned}$$

(2.26)

that is, in the concrete example

$$\begin{aligned} \tau _{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x \in (0,\infty ), \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$

(2.27)

denoting the associated (pre)minimal and maximal operators in $L^2((0,\infty );dx)$ by $T_{2n, min}(c)$, $\overset{{\textbf {.}}}{T}_{2n, min}(c)$, ${\overset{{\textbf {..}}}{T}}_{2n, min}(c)$, $T_{2n,max}(c)$, etc.

In particular, we are interested in the question,

$$\begin{aligned} \begin{aligned}&\text {``for which values of }c \in {{\mathbb {R}}}\text { is }T_{2n, min}(c)\text { self-adjoint} \\&\quad \text {(resp., }{\overset{{\textbf {..}}}{T}}_{2n, min}(c)\text { essentially self-adjoint) in }L^2((0,\infty );dx)?'' \end{aligned} \end{aligned}$$

(2.28)

3 Perturbed Euler Differential Systems and Their Deficiency Indices

In this section we will prove that it suffices to focus on the spectral parameter $\mu = 0$ when trying to determine the number of $L^2((0,d);dx)$-solutions $y(\mu ,\,\cdot \,)$ of

$$\begin{aligned} \begin{aligned} \tau _{2n}(c) y(\mu ,x)&= (-1)^n y^{(2n)}(\mu ,x) + c x^{-2n} y(\mu ,x) = \mu y(\mu ,x), \\&\qquad x \in (0,d), \; \mu \in {{\mathbb {C}}}, \; n \in {{\mathbb {N}}}, \; c \in {{\mathbb {R}}}, \end{aligned} \end{aligned}$$

(3.1)

for fixed $d \in (0,\infty )$ (e.g., one could simply choose $d=1$). In particular, the deficiency indices of the underlying minimal differential operator $T_{2n, min}(c)$ can be determined from the knowledge of the number of $L^2((0,d);dx)$-solutions of $y(0,\,\cdot \,)$, that is, one can reduce (3.1) to the far simpler case $\mu = 0$.

To prove the $\mu$-independence of the number of $L^2((0,d);dx)$-solutions $y(\mu ,\,\cdot \,)$ of (3.1), we find it convenient to employ a bit of the celebrated theory of regular singular points (singular points of the first kind) for first-order systems of differential equations in the complex domain, going back to G. Frobenius [9], L. Fuchs [10,11,12], and L. Sauvage [40,41,42,43,44]. The theory is aptly summarized in a number of treatises, we just mention [3, p. 17–36], [6, p. 108–135], [13, 148–164], [18, p. 70–92], [19, p. 105–131], [20, 21, p. 182–198], [22, p. 342–352], [25, p. 356–372, Ch. XVI], [36, Ch. V], [48, Ch. 4], and [50, 216–235].

In the following $\zeta \in {{\mathbb {C}}}\backslash \{0\}$ (resp., $\zeta \in D(0;R) \backslash \{0\} = \{\zeta \in {{\mathbb {C}}}\,|\, 0< |\zeta | < R\}$ for some fixed $R \in (0,\infty )$) represents the complex analog of $x \in (0,d)$ in (3.1) and we will study first-order systems of differential equations of the particular form

$$\begin{aligned} Y'(\zeta ) = \zeta ^{-1} A(\zeta ) Y(\zeta ), \end{aligned}$$

(3.2)

where $Y(\,\cdot \,)$ represents either an $N \times 1$ solution vector or an $N \times N$ solution matrix, $N \in {{\mathbb {N}}}$, which generally is multi-valued, and $A(\,\cdot \,)$ is an $N \times N$ entire (resp., analytic in D(0; R)) matrix-valued function,

$$\begin{aligned} A(\zeta ) = \sum _{m \in {{\mathbb {N}}}_0} A_m \, \zeta ^m. \end{aligned}$$

(3.3)

The very special structure (at most a first-order pole of the coefficient matrix at $z = 0$) of the right-hand side of (3.2) then leads to a rather special structure of solutions as described in the following.

As a warm up we briefly discuss the pure Euler situation where $A(\,\cdot \,)$ is actually a constant matrix $A_0 \in {{\mathbb {C}}}^{N \times N}$, that is, we consider

$$\begin{aligned} Y'(\zeta ) = \zeta ^{-1} A_0 Y(\zeta ), \end{aligned}$$

(3.4)

with fundamental (typically, many-valued) matrix solutions of the form

$$\begin{aligned} Y(\zeta ) = \zeta ^{A_0} C = e^{A_0 {\text {ln}}(\zeta )} C, \end{aligned}$$

(3.5)

where $C \in {{\mathbb {C}}}^{N \times N}$ is nonsingular (i.e., ${\det }_{{{\mathbb {C}}}^N}(C) \ne 0$). Transforming $A_0$ into its Jordan normal form $\widehat{A}_0 = T A_0 T^{-1}$ for some nonsingular $T \in {{\mathbb {C}}}^{N \times N}$, and setting $\widehat{Y}(\,\cdot \,) = T Y(\,\cdot \,)$ yields

$$\begin{aligned} \widehat{Y}'(\zeta ) = \zeta ^{-1} \widehat{A}_0 \widehat{Y}(\zeta ), \end{aligned}$$

(3.6)

hence one can assume without loss of generality that $A_0$ is in Jordan normal form. In this case $A_0$ is represented as a block diagonal matrix consisting possibly of a diagonal matrix D and possibly of a number of nontrivial Jordan blocks of varying $r \times r$, $1 \le r \le N$, sizes, denoted by $J_r(\alpha _q)$. In particular, if $J_r(\alpha _q)$ is of the form

$$\begin{aligned} J_r(\alpha _q) = \begin{pmatrix} \alpha _q &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{} \quad 0 \\ 0 &{}\quad \alpha _q &{} \quad 1 &{} \quad \cdots &{} \quad 0 \\ \vdots &{} \quad \vdots &{} \quad \ddots &{} \quad \ddots &{} \quad \vdots \\ 0 &{}\quad 0 &{} \quad 0 &{}\quad \cdots &{} \quad 1 \\ 0 &{}\quad 0&{}\quad 0 &{}\quad \cdots &{} \quad \alpha _q \end{pmatrix}, \quad \alpha _q \in \sigma (A_0), \end{aligned}$$

(3.7)

then

$$\begin{aligned} \zeta ^{J_r(\alpha _q)} = \zeta ^{\alpha _q} \begin{pmatrix} 1&{} \quad {\text {ln}}(\zeta ) &{}\quad [{\text {ln}}(\zeta )]^2/[2!] &{} \quad \cdots &{} \quad [{\text {ln}}(\zeta )]^{r-1}/[(r-1)!] \\ 0 &{} \quad 1 &{}\quad {\text {ln}}(\zeta ) &{}\quad \cdots &{}\quad [{\text {ln}}(\zeta )]^{r-2}/[(r-2)!] \\ \vdots &{}\quad \vdots &{}\quad \ddots &{} \quad \ddots &{}\quad \vdots \\ 0 &{} \quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad {\text {ln}}(\zeta ) \\ 0 &{}\quad 0&{}\quad 0 &{}\quad \cdots &{}\quad 1 \end{pmatrix}, \nonumber \\ \end{aligned}$$

(3.8)

explicitly demonstrating the appearance of powers of logarithms of $\zeta$ in (3.5) in the case where $A_0$ has an eigenvalue $\alpha _q$ whose algebraic multiplicity strictly exceeds its geometric one. In particular, the eigenvalues $\alpha _q$ of $A_0$ are determined via the characteristic equation for $A_0$, also called the indicial equation,

$$\begin{aligned} D_N(z) = {\det }_{{{\mathbb {C}}}^N} (z I_N - A_0) = 0, \quad z \in {{\mathbb {C}}}. \end{aligned}$$

(3.9)

The general, or perturbed, Euler case (3.2) leads to analogous results as follows.

Theorem 3.1

(Hille [21], p. 192–198, Kneser [28]). Given the matrix $A(\,\cdot \,) \in {{\mathbb {C}}}^{N \times N}$ in (3.3) entire (resp., analytic in $D(0;R)$ $)$, the perturbed Euler differential system (3.2) has a fundamental set of (generally, multi-valued ) solutions $Y_j \in {{\mathbb {C}}}^{N \times 1}$, $j = 1,\dots , N$, of the form,

$$\begin{aligned} Y_j(\zeta ; q) = \sum _{m \in {{\mathbb {N}}}_0} p_{j,m,q}({\text{ ln }}(\zeta )) \, \zeta ^{m + \alpha _q}, \quad 1 \le j \le N, \end{aligned}$$

(3.10)

where $\alpha _q$ runs through all distinct eigenvalues of $A_0$ (i.e., all elements of $\sigma (A_0)$ $)$, determined via $D_N (\,\cdot \,) = 0$, and $p_{j,m,q} (\,\cdot \,) \in {{\mathbb {C}}}^{N \times 1}$ are polynomials of degree less than or equal to $N-1$. The series in (3.10) converges for $0< |\zeta | < \infty$ (resp., for $0< |\zeta | < R$ $)$.

In this context we also refer to Sections 4.3, 4.4, particularly, Theorem 4.11, in Teschl [48], for a succinct treatment of the Frobenius method for first-order systems with a pole structure as in (3.2).

We also note that a fundamental matrix solution of (3.2) can be obtained in analogy to (3.5) in the pure Euler case. In particular, under the spectral hypothesis that

$$\begin{aligned} \sigma (A_0) \cap \{\sigma (A_0) + {{\mathbb {Z}}}\} = \emptyset , \end{aligned}$$

(3.11)

it was proven by Fuchs [11, 12] (cf. Hille [22, Theorem 9.5.1]) that the perturbed Euler differential system (3.2) has fundamental matrix solutions of the form

$$\begin{aligned} Y(\zeta ) = \sum _{m \in {{\mathbb {N}}}_0} C_m \, \zeta ^{m I_N + A_0} C, \quad C_0 = I_N, \; C_{\ell } \in {{\mathbb {C}}}^{N \times N}, \; \ell \in {{\mathbb {N}}}, \end{aligned}$$

(3.12)

where again $C \in {{\mathbb {C}}}^{N \times N}$ is nonsingular.

The case where the spectral assumption (3.11) on $A_0$ is violated is much more involved¹. What follows is a shortened description of Hille [22, Theorem 9.5.2], a modified version of Frobenius’ method: If (3.11) does not hold, fundamental matrix solutions of the perturbed Euler differential system (3.2) are of the form

$$\begin{aligned} Y(\zeta ) = \sum _{j=0}^M [{\text {ln}}(\zeta )]^j \sum _{m\in {{\mathbb {N}}}_0} C_{m,j} \, \zeta ^{m I_N + A_0} C, \quad C_{0,0} = [M!] I_N, \; C_{m,j} \in {{\mathbb {C}}}^{N \times N}, \nonumber \\ \end{aligned}$$

(3.13)

and once again $C \in {{\mathbb {C}}}^{N \times N}$ is nonsingular. A characterization of M in (3.13) is possible, see, for instance, [22, p. 342–352].

We conclude this overview by specializing the 1st-order $N \times N$ perturbed Euler system (3.2) to the Nth-order scalar case (a special case of which is depicted in (3.1)). Consider the scalar Nth-order differential equation

$$\begin{aligned} y^{(N)}(\zeta ) + b_{N-1}(\zeta ) y^{(N-1)}(\zeta ) + \cdots + b_1(\zeta ) y'(\zeta ) + b_0(\zeta ) y(\zeta ) = 0, \end{aligned}$$

(3.14)

where the coefficients $b_j(\,\cdot \,)$, $0 \le j \le N-1$, are of the form

$$\begin{aligned} b_j(\zeta ) = \zeta ^{j-N} a_j(\zeta ), \quad a_j(\zeta ) = \sum _{m \in {{\mathbb {N}}}_0} a_{j,m} \, \zeta ^m, \end{aligned}$$

(3.15)

with $a_j(\,\cdot \,)$ entire (resp., analytic in D(0; R)). The scalar ODE (3.14) subordinates to the perturbed Euler differential system (3.2) upon identifying $A(\zeta )$ with the $N \times N$ matrix

$$\begin{aligned} \begin{pmatrix} 0 &{} 1 &{} 0 &{} 0 &{} \dots &{} &{} &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} \dots &{} &{} &{} 0 \\ 0 &{} 0 &{} 2 &{} 1 &{} \dots &{} &{} &{} 0 \\ 0 &{} 0 &{} 0 &{} 3 &{} \dots &{} &{} &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \dots &{} &{} &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \dots &{} \ddots &{} \ddots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} &{} &{} 1 \\ - a_0(\zeta ) &{} - a_1(\zeta ) &{} - a_2(\zeta ) &{} - a_3(\zeta ) &{} \dots &{} &{} &{} (N-1) - a_{N-1}(\zeta ) \end{pmatrix} \end{aligned}$$

(3.16)

and identifying $Y(\zeta )$ with $(Y_1(\zeta ), \dots , Y_N(\zeta ))$, where the solutions $Y_j(\,\cdot \,) \in {{\mathbb {C}}}^{N \times 1}$ are given by

$$\begin{aligned} Y_j(\,\cdot \,) = (y_{j,1}(\,\cdot \,), \dots , y_{j,N}(\,\cdot \,))^\top , \quad y_{j,k}(\zeta ) = \zeta ^{k-1} y_j^{(k-1)}(\zeta ), \quad 1 \le j,k \le N, \nonumber \\ \end{aligned}$$

(3.17)

with $y_j(\,\cdot \,)$, $1 \le j \le N$, linearly independent solutions of (3.14). In this scalar context the matrix $A_0 \in {{\mathbb {C}}}^{N \times N}$ in (3.3) is thus of the form

$$\begin{aligned} A_0 = \begin{pmatrix} 0 &{} 1 &{} 0 &{} 0 &{} \dots &{} &{} &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} \dots &{} &{} &{} 0 \\ 0 &{} 0 &{} 2 &{} 1 &{} \dots &{} &{} &{} 0 \\ 0 &{} 0 &{} 0 &{} 3 &{} \dots &{} &{} &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \dots &{} &{} &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \dots &{} \ddots &{} \ddots &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} &{} &{} 1 \\ - a_{0,0} &{} - a_{1,0} &{} - a_{2,0} &{} - a_{3,0} &{} \dots &{} &{} &{} (N-1) - a_{N-1,0} \end{pmatrix} \end{aligned}$$

(3.18)

and hence the eigenvalues $\alpha _q$ of $A_0$ prominently figuring in the solution (3.10) are determined via the indicial equation (3.9), $D_N(\,\cdot \,) = 0$, where

$$\begin{aligned} D_N(z)&= {\det }_{{{\mathbb {C}}}^N} (z I_N - A_0) \nonumber \\&= \sum _{k=0}^N a_{N-k,0} {\left\{ \begin{array}{ll} \prod _{r=1}^{N-k} [z - (r-1)], &{} 0 \le k \le N-1, \\ 1, &{} k=N, \end{array}\right. } \quad a_{N,0} = 1, \; z \in {{\mathbb {C}}}. \end{aligned}$$

(3.19)

Given these results we can return to the half-line differential expression $\tau _{2n}(c)$ in (3.1), the special case of the scalar case (3.14) with $N = 2n$ and (frequently explicitly indicating the c-dependence of the coefficients)

$$\begin{aligned} b_j(\zeta ;c) = 0, \; 1 \le j \le 2n-1, \quad b_0(\zeta ;c) = (-1)^n c \, \zeta ^{-2n} - (-1)^n \mu , \quad \mu \in {{\mathbb {C}}}, \nonumber \\ \end{aligned}$$

(3.20)

equivalently,

$$\begin{aligned} a_j(\zeta ;c) = 0, \quad 1 \le j \le 2n-1, \quad a_0(\zeta ;c) = (-1)^n c - (-1)^n \mu \, \zeta ^{2n}, \quad \mu \in {{\mathbb {C}}}. \nonumber \\ \end{aligned}$$

(3.21)

In this case the indicial equation further reduces to

$$\begin{aligned} D_{2n}(z;c) = \prod _{j=1}^{2n} [z - (j-1)] + (-1)^n c = 0, \quad z \in {{\mathbb {C}}}. \end{aligned}$$

(3.22)

Thus, we can state the following result.

Theorem 3.2

Let $c \in {{\mathbb {R}}}$, $\mu \in {{\mathbb {C}}}$. Then for any $d \in (0,\infty )$, the number of $L^2((0,d); dx)$-solutions of $\tau _{2n}(c) y(\mu ,\,\cdot \,) = \mu y(\mu ;\,\cdot \,)$, denoted by $\#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big )$, is independent of $\mu$. In particular,

$$\begin{aligned} n \le \#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big ) \le 2n. \end{aligned}$$

(3.23)

Moreover, the deficiency indices $n_{\pm }(T_{2n, min}(c))$ (with $T_{2n, min}(c)$ representing the closure of $\tau _{2n}(c) \big |_{C_0^{\infty }((0,\infty ))}$ in $L^2((0,\infty ); dx)$ $)$ equal

$$\begin{aligned} n_{\pm } (T_{2n, min}(c)) = \#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big ) - n. \end{aligned}$$

(3.24)

and hence

$$\begin{aligned} 0 \le n_{\pm } (T_{2n, min}(c)) \le n. \end{aligned}$$

(3.25)

In particular,

$$\begin{aligned} \begin{aligned}&T_{2n, min}(c)\text { is self-adjoint }\big (\text {equivalently, }{\overset{{\textbf {..}}}{T}}_{2n, min}\text { is essentially self-adjoint }\,\big ) \\&\quad \text {in }L^2((0,\infty );dx)\text { if and only if }\ \#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big ) = n. \end{aligned} \end{aligned}$$

(3.26)

Proof

The $\mu$-independence of $\#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big )$ follows from the structure of the solutions $Y_j$ in (3.10), the fact that for each $d \in (0,\infty )$, the power $x^{\alpha }$ lies in $L^2((0,d); dx)$ if and only if ${\text {Re}}(\alpha ) > - 1/2$, independently of the presence of any logarithmic factors, and finally that only the spectrum of $A_0$ determines the powers $\alpha _q$ in (3.10).

Since $c \in {{\mathbb {R}}}$, $\tau _{2n}(c)$ possesses an anti-unitary conjugation operator (effected by complex conjugation of elements in $L^2((0,\infty ); dx)$) and one obtains by (2.17),

$$\begin{aligned} n_+ (T_{2n, min}(c)) = n_- (T_{2n, min}(c)). \end{aligned}$$

(3.27)

Moreover by a special case of Kodaira’s decomposition principle (2.19) for deficiency indices,

$$\begin{aligned} n_{\pm }(T_{2n, min}(c))&= n_{\pm }\Big (\tau _{2n}(c)\big |_{C_0^{\infty }((0,d))}\Big ) + n_{\pm }\Big (\tau _{2n}(c)\big |_{C_0^{\infty }((d,\infty ))}\Big ) - 2n \nonumber \\&= n_{\pm }\Big (\tau _{2n}(c)\big |_{C_0^{\infty }((0,d))}\Big ) - n \nonumber \\&= \#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big ) - n, \end{aligned}$$

(3.28)

since

$$\begin{aligned} n_{\pm }\Big (\tau _{2n}(c)\big |_{C_0^{\infty }((d,\infty ))}\Big ) = n. \end{aligned}$$

(3.29)

Relation (3.29) holds since $\tau _{2n}(c)$ is regular at d and, as $x^{-2n}$ is bounded on the interval $[d,\infty )$ (cf. [34, Sect. 14.7]), $\tau _{2n}(c)$ is in the limit point case at $\infty$ since $(-1)^n d^{2n}/dx^{2n}$ is in the limit point case at $\infty$. Moreover, by (2.18),

$$\begin{aligned} n \le n_{\pm }\Big (\tau _{2n}(c)\big |_{C_0^{\infty }((0,d))}\Big ) \le 2n, \end{aligned}$$

(3.30)

implying (3.23) and (3.25). $\square$

Remark 3.3

(i) The independence of $\#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big )$ with respect to $\mu$ permits one to choose the by far simplest situation by taking $\mu = 0$ when counting the number of $L^2((0,d); dx)$-solutions of $\tau _{2n}(c) y(\mu ,\,\cdot \,) = \mu y(\mu ;\,\cdot \,)$. This in turn grants one to focus on solutions of the simple power-type $x^{\alpha }$ as in (3.10) (ignoring the possibility of additional logarithmic factors which, however, cannot influence the $L^2$- or non-$L^2$-behavior of solutions near $x=0$). In particular, considering

$$\begin{aligned} y_{\alpha }(x) = x^{\alpha } P({\text {ln}}(x)), \quad x \in (0,\infty ), \; \alpha \in {{\mathbb {C}}}, \end{aligned}$$

(3.31)

where $P(\,\cdot \,)$ is any polynomial, then for all $d \in (0,\infty )$,

$$\begin{aligned} y_{\alpha }(\,\cdot \,) \in L^2((0,d); dx) \, \text { if and only if } \, {\text {Re}}(\alpha ) > -1/2. \end{aligned}$$

(3.32)

Thus, by (3.10), ${\text {Re}}(\alpha ) > - 1/2$, respectively, ${\text {Re}}(\alpha ) \le - 1/2$, is the criterion deciding whether or not a particular solution with power-type behavior $x^{\alpha }$ (again, ignoring possible logarithmic factors) contributes to $\#_{L^2}\big (\tau _{2n}(c)|_{(0,d)}\big )$.

(ii) It will be shown in Corollary 4.8 that any permissible integer value for $\#(\tau _{2n}|_{(0,d)})$ in (3.23) actually is attained for some $c \in {{\mathbb {R}}}$. $\diamond$

Remark 3.4

One observes that $D_{2n}(\,\cdot \,;c)$ possesses the symmetry

$$\begin{aligned} D_{2n}(-(1/2) + n + z) = D_{2n}(-(1/2) + n - z). \end{aligned}$$

(3.33)

In particular, at $z=0$ one obtains

$$\begin{aligned} D_{2n}((-1/2) + n)= & {} (-1)^n \Bigg ( \prod _{j=1}^{n} [j - 1/2]^2 + c\Bigg ) \nonumber \\= & {} (-1)^n \left( \frac{[(2n -1)!!]^{2}}{2^{2n}} + c \right) . \end{aligned}$$

(3.34)

Consequently, for $c=-[(2n -1)!!]^{2}\big /2^{2n}$ one has a double zero at $\alpha =k - (1/2)$ and there are two solutions of the type

$$\begin{aligned} y_1(0,x,c) = x^{k-(1/2)}, \qquad y_2(0,x,c) = x^{k-(1/2)} {\text {ln}}(x) \end{aligned}$$

(3.35)

in this case. $\diamond$

Next, we now recall the special situation $n=1$ which is explicitly solvable for general spectral parameter $\mu$ in terms of Bessel functions as follows:

Example 3.5

Assuming the case $n=1$ in (3.1) we consider

$$\begin{aligned} \begin{aligned}&- y''(\mu ,x) + c x^{-2} y(\mu ,x) = \mu y(\mu ,x), \\&\quad \mu \in {{\mathbb {C}}}, \; x \in (0,\infty ), \; c \in {{\mathbb {R}}}. \end{aligned} \end{aligned}$$

(3.36)

The associated characteristic equation

$$\begin{aligned} D_2(z; c) = z(z - 1) - c = 0, \end{aligned}$$

(3.37)

has the following two complex-valued solutions

$$\begin{aligned} \begin{aligned} \alpha _1(c)&= (1/2) - \sqrt{c + (1/4)}, \\ \alpha _2(c)&= (1/2) + \sqrt{c + (1/4)}, \end{aligned} \end{aligned}$$

(3.38)

choosing the principal branch for $[ \,\cdot \,]^{1/2}$ with branch cut $(-\infty ,0]$, such that

$$\begin{aligned} z^{1/2} = r^{1/2} e^{i \varphi /2}, \quad z = r e^{i \varphi }, \quad r, r^{1/2} \in [0,\infty ), \; \varphi \in (-\pi ,\pi ]. \end{aligned}$$

(3.39)

With this convention in place one checks that for all $c \in {{\mathbb {R}}}$, one has the ordering,

$$\begin{aligned} {\text {Re}}(\alpha _1(c)) \le 1/2 \le {\text {Re}}(\alpha _2(c)). \end{aligned}$$

(3.40)

$(\alpha )$ Generic case: Suppose $c\in {\mathbb {R}}$ is such that

$$\begin{aligned} {[}\alpha _1(c)-\alpha _2(c)]/2 \not \in {\mathbb {Z}}. \end{aligned}$$

(3.41)

Then the nonhomogenous differential equation (3.36) has the following fundamental system of solutions (cf. [1, No. 9.1.49, p. 362])

$$\begin{aligned} y_1(\mu ,x; c)&= (\pi /2) \mu ^{-\gamma (c)/2} x^{1/2} J_{\gamma (c)} \big (\mu ^{1/2}x\big ), \nonumber \\ y_2(\mu ,x; c)&= \sin (\pi \gamma (c)) \mu ^{\gamma (c)/2} x^{1/2} J_{-\gamma (c)} \big (\mu ^{1/2}x\big ), \nonumber \\ \mu&\in {{\mathbb {C}}}, \; x\in (0,\infty ), \end{aligned}$$

(3.42)

where

$$\begin{aligned} \gamma (c) = \sqrt{c + (1/4)}, \quad \gamma \in [0,\infty ), \quad c \in {{\mathbb {R}}}. \end{aligned}$$

(3.43)

(Thus, $\gamma (c) \in \{[0,\infty ) \backslash {{\mathbb {N}}}_0\} \cup i (0,\infty )$ in the generic case.)

$(\beta )$ Exceptional Cases: Suppose $c \in {{\mathbb {R}}}$ is such that

$$\begin{aligned}{}[\alpha _1(c)-\alpha _2(c)]/2 \in {\mathbb {Z}}, \end{aligned}$$

(3.44)

then

$$\begin{aligned} c = k^2 - (1/4), \quad k \in {{\mathbb {N}}}_0. \end{aligned}$$

(3.45)

More precisely, for $k\in {{\mathbb {N}}}_0$,

$$\begin{aligned}{}[\alpha _{1}(c)-\alpha _{2}(c)]/2 = \pm k \, \text { if and only if } \, c=k^2 - (1/4). \end{aligned}$$

(3.46)

Furthermore,

$$\begin{aligned} \alpha _1(c)= \alpha _2(c) \, \text { if and only if } \, c = - 1/4. \end{aligned}$$

(3.47)

In the exceptional case, where $\gamma (c) = k \in {{\mathbb {N}}}_0$, one obtains

$$\begin{aligned} y_1\big (\mu ,x; k^2-(1/2)\big )&= (\pi /2) \mu ^{-k/2} x^{1/2} J_k \big (\mu ^{1/2}x\big ), \nonumber \\ y_2\big (\mu ,x; k^2-(1/2)\big )&= \mu ^{k/2} x^{1/2} \big [ -Y_k \big (\mu ^{1/2}x\big ) + \pi ^{-1}{\text {ln}}(\mu )J_k \big (\mu ^{1/2}x\big )\big ], \nonumber \\ \mu&\in {{\mathbb {C}}}, \; x\in (0,\infty ), \; c \in \big \{k^2-(1/4)\big \}_{k\in {{\mathbb {N}}}_0}. \end{aligned}$$

(3.48)

Here $J_{\kappa }(\,\cdot \,)$ represent the standard Bessel functions of order $\kappa \in {{\mathbb {C}}}$ and first kind, and $Y_k(\,\cdot \,)$ denotes the Bessel function of order $k \in {{\mathbb {N}}}_0$ and second kind (see, e.g., [1, Ch. 9]). Moreover, one verifies (cf. [1, p. 360]) that

$$\begin{aligned} W(y_2(\mu ,\,\cdot \,, c), y_1(\mu ,\,\cdot \,; c)) = 1, \quad \mu \in {{\mathbb {C}}}, \; c \in {{\mathbb {R}}} \end{aligned}$$

(3.49)

(here $W(f,g) = fg' - f'g$ denotes the Wronskian of f and $g$ $)$, and that the fundamental system of solutions $y_1(\mu ,\cdot ; c), y_2(\mu ,\cdot , c)$ (3.42), (3.48) of (3.36) is entire with respect to $\mu \in {{\mathbb {C}}}$ for fixed $x \in (0,\infty )$, and real-valued for $\mu \in {{\mathbb {R}}}$.

As $\mu \rightarrow 0$, the fundamental systems of solutions (3.42), (3.48), upon disregarding normalization, greatly simplify to

$$\begin{aligned} y_1(0,x; c)= x^{\alpha _1(c)}, \quad c \in {{\mathbb {R}}}, \quad y_2(0,x; c) =&{\left\{ \begin{array}{ll} x^{\alpha _2(c)}, \quad c \in {{\mathbb {R}}}\backslash \{- 1/4\}, \\ x^{1/2} {\text{ ln }}(x), \quad c = -1/4; \end{array}\right. } \nonumber \\ {}&\qquad x\in (0,\infty ), \end{aligned}$$

(3.50)

underscoring once again the advantage of choosing $\mu = 0$.

One observes that in accordance with (1.9) (see also (1.10)) and Remark 3.4, the logarithmic case in (3.50) occurs at $c = -1/4$, that is, precisely at the borderline of semiboundedness of $T_{min,2}(c)$.

Thus, determining whether or not ${\text {Re}}(\alpha _j(c) > - 1/2$, $j=1,2$, one concludes that

$$\begin{aligned} \#_{L^2}\big (\tau _{2}(c)|_{(0,d)}\big ) = {\left\{ \begin{array}{ll} 1, &{} \text{ if }\quad c \ge 3/4, \\ 2, &{} \text{ if }\quad c < 3/4. \end{array}\right. } \end{aligned}$$

(3.51)

Remark 3.6

In view of the next example, where $n=2$, in fact, in view of the general case $n \in {{\mathbb {N}}}$, it might be interesting to rewrite the Bessel function solutions in the case $n=1$ in terms of the corresponding generalized hypergeometric function and Meijer’s G-function as follows: In the generic case, where $c\in {\mathbb {R}}$ is such that $[\alpha _1(c)-\alpha _2(c)]/2 \not \in {\mathbb {Z}}$, the nonhomogenous differential equation (3.36) has the following fundamental system of solutions

$$\begin{aligned} y_1(\mu ,x; c)&= x^{\alpha _1(c)} \,_{0} F_1\left( \!\begin{array}{c}\\ {\scriptstyle 1+\frac{\alpha _1(c)-\alpha _2(c)}{2}} \end{array} \bigg \vert \, -\frac{\mu x^2}{4} \right) , \nonumber \\ y_2(\mu ,x; c)&= x^{\alpha _2(c)} \,_{0} F_1\left( \!\begin{array}{c}\\ {\scriptstyle 1+\frac{\alpha _2(c)-\alpha _1(c)}{2}} \end{array} \bigg \vert \, -\frac{\mu x^2}{4} \right) , \nonumber \\ \mu&\in {{\mathbb {C}}}, \; x\in (0,\infty ). \end{aligned}$$

(3.52)

Here $\,_{0} F_1\Big (\!\begin{array}{c}\\ {\scriptstyle b_1} \end{array} \Big \vert \, \,\cdot \,\Big )$ represents the generalized hypergeometric function given by

$$\begin{aligned} \,_{0} F_1\Big (\!\begin{array}{c}\\ {\scriptstyle b_1} \end{array} \Big \vert \, \zeta \Big ) = \sum _{k \in {{\mathbb {N}}}_0} \frac{\zeta ^k}{(b_1)_k k!}, \quad b_1 \in {{\mathbb {C}}}\backslash \{- {{\mathbb {N}}}_0\}, \; \zeta \in {{\mathbb {C}}}, \end{aligned}$$

(3.53)

with $(a)_k$ denoting Pochhammer’s symbol,

$$\begin{aligned} (a)_0 =1, \quad (a)_k = \prod _{j=0}^{k-1} (a+j) = \Gamma (a+k)/\Gamma (a), \quad k \in {{\mathbb {N}}}, \; a \in {{\mathbb {C}}}. \end{aligned}$$

(3.54)

In particular, $\,_{0} F_1\Big (\!\begin{array}{c}\\ {\scriptstyle b_1} \end{array} \Big \vert \, \zeta \Big )$ is entire in $\zeta \in {{\mathbb {C}}}$ and

$$\begin{aligned} \,_{0} F_1\Big (\!\begin{array}{c}\\ {\scriptstyle b_1} \end{array} \Big \vert \, \zeta \Big ) \underset{\zeta \rightarrow 0}{=}\ 1 + O(\zeta ). \end{aligned}$$

(3.55)

In the exceptional case, where $\gamma (c) = k \in {{\mathbb {N}}}_0$, one obtains

$$\begin{aligned} y_1\big (\mu ,x; k^2-(1/2)\big )&= x^{k+(1/2)} \,_{0} F_1\left( \!\begin{array}{c}\\ {\scriptstyle 1+k} \end{array} \bigg \vert \, -\frac{\mu x^2}{4} \right) , \nonumber \\ y_2\big (\mu ,x; k^2-(1/2)\big )&= \Gamma (k+1) 2^k \mu ^{-k/2} x^{1/2} G_{0,2}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle k/2 ; -k/2} \end{array} \bigg \vert \, - \frac{\mu x^2}{4}\right) \nonumber \\&\quad + \big [\pi (-1)^{k+1} i^{k+1} + {\text {ln}}(\mu )\big ] x^{k+(1/2)} \,_{0} F_1\left( \!\begin{array}{c}\\ {\scriptstyle 1+k} \end{array} \bigg \vert \, -\frac{\mu x^2}{4} \right) , \nonumber \\&\quad \mu \in {{\mathbb {C}}}, \; x\in (0,\infty ), \; c \in \big \{k^2-(1/4)\big \}_{k\in {{\mathbb {N}}}_0}. \end{aligned}$$

(3.56)

Here Meijer’s G-function, $G_{0,2}^{2,0}\Big (\!\begin{array}{c}\\ {\scriptstyle c_1, c_2} \end{array} \Big \vert \, \,\cdot \,\Big )$, is given by a Mellin–Barnes-type integral,

$$\begin{aligned} G_{0,2}^{2,0}\Big (\!\begin{array}{c}\\ {\scriptstyle c_1, c_2} \end{array} \Big \vert \, \zeta \Big ) = \frac{1}{2\pi i} \int _{{{\mathcal {C}}}} ds \, \zeta ^s \Gamma (c_1-s) \Gamma (c_2-s), \end{aligned}$$

(3.57)

where ${{\mathcal {C}}}$ is a contour beginning and ending at $+\infty$ encircling all poles of $\Gamma (c_j-s)$, $j=1,2$, once in negative orientation, and the left-hand side of (3.57) is defined as the (absolutely convergent) sum of residues of the right-hand side. The exceptional case where $c_1$ and $c_2$ differ by an integer is treated by a limiting argument. (For more details, see [14].) $\diamond$

For details on generalized hypergeometric functions and Meijer’s G-function we refer, for instance, to [4, 8, Ch. IV, Sects. 5.3–5.6], [31, Ch. V], [32, Ch. V], and [35, Ch. 16], [37, Sect. 8.2].

Example 3.7

Assuming the case $n=2$ in (3.1) we consider

$$\begin{aligned} \begin{aligned}&y''''(\mu ,x) + c x^{-4} y(\mu ,x) = \mu y(\mu ,x), \\&x \in (0,\infty ), \; \mu \in {{\mathbb {C}}}, \; c \in {{\mathbb {R}}}. \end{aligned} \end{aligned}$$

(3.58)

The associated characteristic equation

$$\begin{aligned} D_4(z; c) = z(z - 1)(z-2)(z-3) - c = 0, \quad z \in {{\mathbb {C}}}, \; c \in {{\mathbb {R}}}, \end{aligned}$$

(3.59)

has the following four complex-valued solutions,

$$\begin{aligned} \begin{aligned} \alpha _1(c)&= \left[ 3-\sqrt{5+4 \sqrt{1-c}} \,\right] \bigg /2,\\ \alpha _2(c)&= \left[ 3-\sqrt{5-4 \sqrt{1-c}} \,\right] \bigg /2,\\ \alpha _3(c)&= \left[ 3+\sqrt{5-4 \sqrt{1-c}} \,\right] \bigg /2,\\ \alpha _4(c)&= \left[ 3+\sqrt{5+4 \sqrt{1-c}} \, \right] \bigg /2; \quad c \in {{\mathbb {R}}}, \end{aligned} \end{aligned}$$

(3.60)

employing the principal branch (3.39) for $[\,\cdot \,]^{1/2}$. With this convention, one checks that for all $c\in {\mathbb {R}}$, one has

$$\begin{aligned} {\text {Re}}(\alpha _1(c)) \le {\text {Re}}(\alpha _2(c)) \le 3/2 \le {\text {Re}}(\alpha _3(c)) \le {\text {Re}}(\alpha _4(c)). \end{aligned}$$

(3.61)

$(\alpha )$ Generic case: Suppose $c\in {\mathbb {R}}$ is such that

$$\begin{aligned} {}[\alpha _{j}(c)-\alpha _{j'}(c)]/4 \not \in {\mathbb {Z}}, \, \text { for all } \, 1 \le j, j' \le 4, \; j\not =j'. \end{aligned}$$

(3.62)

Then the nonhomogenous differential equation (3.58) has the following fundamental system of solutions,

$$\begin{aligned} \begin{aligned} y_1(\mu ,x;c)&=x^{\alpha _1(c)} \,_{0} F_3\left( \!\begin{array}{c}\\ {\scriptstyle 1+\frac{\alpha _1(c)-\alpha _2(c)}{4}, 1+\frac{\alpha _1(c)-\alpha _3(c)}{4}, 1+\frac{\alpha _1(c)-\alpha _4(c)}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_2(\mu ,x;c)&=x^{\alpha _2(c)} \,_{0} F_3\left( \!\begin{array}{c}\\ {\scriptstyle 1+\frac{\alpha _2(c)-\alpha _1(c)}{4}, 1+\frac{\alpha _2(c)-\alpha _3(c)}{4}, 1+\frac{\alpha _2(c)-\alpha _4(c)}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_3(\mu ,x;c)&=x^{\alpha _3(c)} \,_{0} F_3\left( \!\begin{array}{c}\\ {\scriptstyle 1+\frac{\alpha _3(c)-\alpha _1(c)}{4}, 1+\frac{\alpha _3(c)-\alpha _2(c)}{4}, 1+\frac{\alpha _3(c)-\alpha _4(c)}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_4(\mu ,x;c)&=x^{\alpha _4(c)} \,_{0} F_3\left( \!\begin{array}{c}\\ {\scriptstyle 1+\frac{\alpha _4(c)-\alpha _1(c)}{4}, 1+\frac{\alpha _4(c)-\alpha _2(c)}{4}, 1+\frac{\alpha _4(c)-\alpha _3(c)}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ; \\ {}&\qquad \mu \in {{\mathbb {C}}}, \; x \in (0,\infty ). \end{aligned} \end{aligned}$$

(3.63)

Asymptotically,

$$\begin{aligned} y_j(\mu ,x;c) \underset{x\downarrow 0}{=}\ x^{\alpha _j(c)}[1+O(x)], \quad 1 \le j \le 4, \end{aligned}$$

(3.64)

and thus, the four functions are indeed linearly independent.

Here $\,_{0} F_3\Big (\!\begin{array}{c}\\ {\scriptstyle b_1, b_2, b_3} \end{array} \Big \vert \, \,\cdot \,\Big )$ represents the generalized hypergeometric function given by

$$\begin{aligned} \,_{0} F_3\Big (\!\begin{array}{c}\\ {\scriptstyle b_1, b_2, b_3} \end{array} \Big \vert \, \zeta \Big ) = \sum _{k \in {{\mathbb {N}}}_0} \frac{\zeta ^k}{(b_1)_k (b_2)_k (b_3)_k k!}, \quad b_1, b_2, b_3 \in {{\mathbb {C}}}\backslash \{- {{\mathbb {N}}}_0\}, \; \zeta \in {{\mathbb {C}}}. \end{aligned}$$

(3.65)

Again, $\,_{0} F_3\Big (\!\begin{array}{c}\\ {\scriptstyle b_1, b_2, b_3} \end{array} \Big \vert \, \zeta \Big )$ is entire in $\zeta \in {{\mathbb {C}}}$ and

$$\begin{aligned} \,_{0} F_3\Big (\!\begin{array}{c}\\ {\scriptstyle b_1, b_2, b_3} \end{array} \Big \vert \, \zeta \Big ) \underset{\zeta \rightarrow 0}{=}\ 1 + O(\zeta ). \end{aligned}$$

(3.66)

That these functions are in fact solutions of (3.58) can be confirmed by direct verification using the differential equation for generalized hypergeometric functions.

$(\beta )$ Exceptional Cases: Suppose $c \in {{\mathbb {R}}}$ is such that

$$\begin{aligned}{}[\alpha _{j}(c)-\alpha _{j'}(c)]/4 \in {\mathbb {Z}} \, \text { for some } \, 1 \le j, j' \le 4, \; j\not =j', \end{aligned}$$

(3.67)

then

$$\begin{aligned} \text {either } \, c = 1 - 20 k^2 + 64 k^4, \, \text { or, } \, c = - (9/16) +10 k^2 - 16 k^4, \quad k \in {{\mathbb {N}}}_0. \nonumber \\ \end{aligned}$$

(3.68)

More precisely, for $k\in {{\mathbb {N}}}_0$,

$$\begin{aligned} \begin{aligned}{}[\alpha _{1}(c)-\alpha _{2}(c)]/4&= \pm k \, \text{ implies } \, c=1 - 20 k^2 + 64 k^4,\\ [\alpha _{1}(c)-\alpha _{3}(c)]/4&= \pm k \, \text{ implies } \, c=1 - 20 k^2 + 64 k^4,\\ [\alpha _{1}(c)-\alpha _{4}(c)]/4&= \pm k \, \text{ implies } \, c= - (9/16) +10 k^2 - 16 k^4,\\ [\alpha _{2}(c)-\alpha _{3}(c)]/4&= \pm k \, \text{ implies } \, c= - (9/16) +10 k^2 - 16 k^4,\\ [\alpha _{2}(c)-\alpha _{4}(c)]/4&= \pm k \, \text{ implies } \, c=1 - 20 k^2 + 64 k^4,\\ [\alpha _{3}(c)-\alpha _{4}(c)]/4&= \pm k \, \text{ implies } \, c=1 - 20 k^2 + 64 k^4. \end{aligned} \end{aligned}$$

(3.69)

Furthermore,

$$\begin{aligned} \alpha _1(c)= \alpha _2(c) \, \text { if and only if } \, \alpha _3(c)= \alpha _4(c) \, \text { if and only if } \, c =1 \end{aligned}$$

(3.70)

and

$$\begin{aligned} \alpha _2(c)= \alpha _3(c) \, \text { if and only if } \, c = - 9/16. \end{aligned}$$

(3.71)

If $c=1$, then

$$\begin{aligned} \alpha _1(1) = \alpha _2(1) = \big [3 - \sqrt{5}\,\big ]\big /2, \quad \alpha _3(1) = \alpha _4(1) = \big [3 + \sqrt{5}\,\big ]\big /2, \end{aligned}$$

(3.72)

and a fundamental system of solutions is given by,

$$\begin{aligned} \begin{aligned} y_1(\mu ,x;1)&=x^{[3-\sqrt{5}]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1, 1-\frac{\sqrt{5}}{4},1-\frac{\sqrt{5}}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_2(\mu ,x;1)&=G_{0,4}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{3-\sqrt{5}}{8} ,\frac{3-\sqrt{5}}{8} ; \frac{3+\sqrt{5}}{8},\frac{3+\sqrt{5}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_3(\mu ,x;1)&=x^{[3+\sqrt{5}]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1, 1+\frac{\sqrt{5}}{4},1+\frac{\sqrt{5}}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_4(\mu ,x;1)&=G_{0,4}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{3+\sqrt{5}}{8} ,\frac{3+\sqrt{5}}{8} ; \frac{3-\sqrt{5}}{8},\frac{3-\sqrt{5}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ; \\&\qquad \mu \in {{\mathbb {C}}}, \; x \in (0,\infty ). \end{aligned} \end{aligned}$$

(3.73)

Asymptotically,

$$\begin{aligned} \begin{aligned}&y_2(\mu ,x;1) \underset{x \downarrow 0}{=}\ c_2 x^{[3-\sqrt{5}]/2} {\text {ln}}(x)[1 + O(x)], \\&y_4(\mu ,x;1) \underset{x \downarrow 0}{=}\ c_4 x^{[3+\sqrt{5}]/2} {\text {ln}}(x)[1 + O(x)]. \end{aligned} \end{aligned}$$

(3.74)

Here Meijer’s G-function, $G_{0,4}^{2,0}\Big (\!\begin{array}{c}\\ {\scriptstyle c_1, c_2; c_3, c_4} \end{array} \Big \vert \, \,\cdot \,\Big )$, is again given by a Mellin–Barnes-type integral,

$$\begin{aligned} G_{0,4}^{2,0}\Big (\!\begin{array}{c}\\ {\scriptstyle c_1, c_2; c_3, c_4} \end{array} \Big \vert \, \zeta \Big ) = \frac{1}{2\pi i} \int _{{{\mathcal {C}}}} ds \, \zeta ^s \frac{\Gamma (c_1-s) \Gamma (c_2-s)}{\Gamma (1-c_3+s) \Gamma (1-c_4+s)}, \end{aligned}$$

(3.75)

where ${{\mathcal {C}}}$ is a contour beginning and ending at $+\infty$ encircling all poles of $\Gamma (c_j-\cdot )$, $j=1,2$, once in negative orientation, and the left-hand side of (3.75) is defined as the (absolutely convergent) sum of residues of the right-hand side. The exceptional case where $c_1$ and $c_2$ differ by an integer is once more treated by a limiting argument.

If $c = 1 - 20 k^2 + 64 k^4$, $k\in {{\mathbb {N}}}$, then

$$\begin{aligned} \begin{aligned} \alpha _1\big (1 - 20 k^2 + 64 k^4\big )&= \Big [3 - 4k - \sqrt{5 - 16k^2} \,\Big ]\Big /2, \\ \alpha _2\big (1 - 20 k^2 + 64 k^4\big )&= \Big [3 - 4k + \sqrt{5 - 16k^2} \,\Big ]\Big /2, \\ \alpha _3\big (1 - 20 k^2 + 64 k^4\big )&= \Big [3 + 4k - \sqrt{5 - 16k^2} \,\Big ]\Big /2, \\ \alpha _4\big (1 - 20 k^2 + 64 k^4\big )&= \Big [3 + 4k + \sqrt{5 - 16k^2} \,\Big ]\Big /2, \\ \end{aligned} \end{aligned}$$

(3.76)

and a fundamental system of solutions is given by,

$$\begin{aligned}&y_1\big (\mu ,x; 1 - 20 k^2 + 64 k^4\big ) \nonumber \\ {}&\quad =G_{0,4}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{3-4 k-\sqrt{5-16 k^2}}{8} , \frac{3+4 k-\sqrt{5-16 k^2}}{8} ; \frac{3-4 k+\sqrt{5-16 k^2}}{8} , \frac{3+4 k+\sqrt{5-16 k^2}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256}\right) , \nonumber \\ {}&y_2\big (\mu ,x; 1 - 20 k^2 + 64 k^4\big ) \nonumber \\ {}&\quad =G_{0,4}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{3-4 k+\sqrt{5-16 k^2}}{8} , \frac{3+4 k+\sqrt{5-16 k^2}}{8} ; \frac{3-4 k-\sqrt{5-16 k^2}}{8} , \frac{3+4 k-\sqrt{5-16 k^2}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256}\right) , \nonumber \\ {}&y_3\big (\mu ,x; 1 - 20 k^2 + 64 k^4\big ) \nonumber \\ {}&\quad = x^{[(3+4k) - \sqrt{5-16k^2}\,]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1 + k , 1 + k - \frac{\sqrt{5 - 16 k^2}}{4},1 - \frac{\sqrt{5 - 16 k^2}}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) , \nonumber \\ {}&y_4\big (\mu ,x; 1 - 20 k^2 + 64 k^4\big ) \nonumber \\ {}&\quad = x^{[(3+4k) + \sqrt{5-16k^2}\,]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1+k, 1 + k + \frac{\sqrt{5 - 16 k^2}}{4}, 1 + \frac{\sqrt{5 - 16 k^2}}{4}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ; \nonumber \\ {}&\qquad \mu \in {{\mathbb {C}}}, \; x \in (0,\infty ). \end{aligned}$$

(3.77)

Asymptotically,

$$\begin{aligned} \begin{aligned}&y_1\big (\mu ,x; 1 - 20 k^2 + 64 k^4\big ) \underset{x \downarrow 0}{=} x^{[(3-4k) - \sqrt{5-16k^2}\,]/2} {\text {ln}}(x) [1 + O(x)], \\&y_2\big (\mu ,x; 1 - 20 k^2 + 64 k^4\big ) \underset{x \downarrow 0}{=}\ x^{[(3-4k) + \sqrt{5-16k^2}\,]/2} {\text {ln}}(x) [1 + O(x)]. \end{aligned} \end{aligned}$$

(3.78)

If $c=-9/16$, then

$$\begin{aligned} \begin{aligned} \alpha _1(-9/16)&= \big [3 - \sqrt{10}\,\big ]\big /2, \\ \alpha _2(-9/16)&=\alpha _3(-9/16) = 3/2, \\ \alpha _4(-9/16)&= \big [3 + \sqrt{10}\,\big ]\big /2, \end{aligned} \end{aligned}$$

(3.79)

and a fundamental system of solutions is given by,

$$\begin{aligned} \begin{aligned} y_1(\mu ,x;-9/16)&=x^{[3-\sqrt{10}]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1-\frac{\sqrt{10}}{4}, 1-\frac{\sqrt{10}}{8},1-\frac{\sqrt{10}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_2(\mu ,x;-9/16)&=x^{3/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1, 1-\frac{\sqrt{10}}{8},1+\frac{\sqrt{10}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_3(\mu ,x;-9/16)&=G_{0,4}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{3}{8} ,\frac{3}{8} ; \frac{3-\sqrt{10}}{8},\frac{3+\sqrt{10}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ,\\ y_4(\mu ,x;-9/16)&=x^{[3+\sqrt{10}]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1+\frac{\sqrt{10}}{4}, 1+\frac{\sqrt{10}}{8},1+\frac{\sqrt{10}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ; \\ {}&\qquad \mu \in {{\mathbb {C}}}, \; x \in (0,\infty ). \end{aligned} \end{aligned}$$

(3.80)

Asymptotically,

$$\begin{aligned} y_3(\mu ,x, -9/16) \underset{x \downarrow 0}{=}\ c_3 x^{3/2} {\text {ln}}(x) [1 + O(x)]. \end{aligned}$$

(3.81)

One observes that the case $c = -9/16$ is once more precisely the borderline of semiboundedness of $T_{min,4}(c)$ again in accordance with (1.9) (see also (1.10)) and Remark 3.4.

If $c=- (9/16) +10 k^2 - 16 k^4$, $k\in {{\mathbb {N}}}$, then

$$\begin{aligned} \begin{aligned} \alpha _1\big (-(9/16) +10 k^2 - 16 k^4\big )&= (3-4k)/2, \\ \alpha _2\big (-(9/16) +10 k^2 - 16 k^4\big )&= \Big [3 - \sqrt{10-16k^2} \,\Big ]\Big /2, \\ \alpha _3\big (-(9/16) +10 k^2 - 16 k^4\big )&= \Big [3 + \sqrt{10-16k^2} \,\Big ]\Big /2, \\ \alpha _4\big (-(9/16) +10 k^2 - 16 k^4\big )&= (3+4k)/2, \end{aligned} \end{aligned}$$

(3.82)

and a fundamental system of solutions is given by,

$$\begin{aligned}&y_1\big (\mu ,x;-(9/16) +10 k^2 - 16 k^4\big ) \nonumber \\ {}&\quad =G_{0,4}^{2,0}\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{3-4k}{8} , \frac{3+4 k}{8} ; \frac{3-\sqrt{10-16 k^2}}{8} , \frac{3+\sqrt{10-16 k^2}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) , \nonumber \\ {}&y_2\big (\mu ,x;-(9/16) +10 k^2 - 16 k^4\big ) \nonumber \\ {}&\quad =x^{[3 - \sqrt{10-16k^2}\,]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{8-2\sqrt{10 - 16 k^2}}{8}, \frac{8-4k-\sqrt{10 - 16 k^2}}{8}, \frac{8+4k-\sqrt{10 - 16 k^2}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) , \nonumber \\ {}&y_3\big (\mu ,x;-(9/16) +10 k^2 - 16 k^4\big ) \nonumber \\ {}&\quad =x^{[3 + \sqrt{10-16k^2}\,]/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle \frac{8+2\sqrt{10 - 16 k^2}}{8}, \frac{8-4k+\sqrt{10 - 16 k^2}}{8}, \frac{8+4k+\sqrt{10 - 16 k^2}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) , \nonumber \\ {}&y_4\big (\mu ,x;-(9/16) +10 k^2 - 16 k^4\big ) \nonumber \\ {}&\quad =x^{(3+4k)/2} \,_{0} F_3\left( \!\!\begin{array}{c}\\ {\scriptstyle 1+k, \frac{8+4k-\sqrt{10 - 16 k^2}}{8}, \frac{8+4k+\sqrt{10 - 16 k^2}}{8}} \end{array} \bigg \vert \, \frac{\mu x^4}{256} \right) ; \nonumber \\ {}&\qquad \mu \in {{\mathbb {C}}}, \; x \in (0,\infty ). \end{aligned}$$

(3.83)

Asymptotically,

$$\begin{aligned} y_1(x) \underset{x \downarrow 0}{=}\ c_1 x^{(3-4k)/2} [1 + O(x)] + c_2 x^{(3+4k)/2} {\text {ln}}(x) [1 + O(x)]. \end{aligned}$$

(3.84)

Once more, as $\mu \rightarrow 0$, the fundamental system of solutions of (3.58) considerably simplifies to

$$\begin{aligned} y_1(0,x;c)&= x^{\alpha _1(c)}, \quad y_2(0,x;c) = x^{\alpha _2(c)}, \nonumber \\ y_3(0,x;c)&= x^{\alpha _3(c)}, \quad y_4(0,x;c) = x^{\alpha _4(c)}; \quad c \in {{\mathbb {R}}}\backslash \{1, - 9/16\}, \end{aligned}$$

(3.85)

$$\begin{aligned} y_1(0,x;1)&= x^{[3-\sqrt{5}]/2}, \quad y_2(0,x;1) = x^{[3-\sqrt{5}]/2} {\text {ln}}(x), \nonumber \\ y_3(0,x;1)&= x^{[3+\sqrt{5}]/2}, \quad y_4(0,x;1) = x^{[3+\sqrt{5}]/2} {\text {ln}}(x), \quad c=1, \end{aligned}$$

(3.86)

$$\begin{aligned} y_1(0,x;-9/16)&= x^{[3-\sqrt{10}]/2}, \quad y_3(0,x;-9/16) = x^{3/2}, \nonumber \\ y_3(0,x;-9/16)&= x^{3/2} {\text {ln}}(x), \quad y_4(0,x;-9/16) = x^{[3+\sqrt{10}]/2}, \quad c=-9/16; \nonumber \\ x&\in (0,\infty ). \end{aligned}$$

(3.87)

By inspection, one verifies that $\tau _4(c) y_j(0,\,\cdot \,;c) =0$, $1 \le j \le 4$. Alternatively, one can apply the theory of nth-order Euler differential equations as presented, for instance, in [6, p. 122–123].

Thus, determining whether or not ${\text {Re}}(\alpha _j(c) > - 1/2$, $1 \le j \le 4$, one concludes that

$$\begin{aligned} \#_{L^2}\big (\tau _{4}(c)|_{(0,d)}\big ) = {\left\{ \begin{array}{ll} 2, &{} \text{ if }\quad c \ge 45, \\ 4, &{} \text{ if }\quad - (7!!)/2^4 \le c< 45, \\ 3, &{} \text{ if }\quad c < - (7!!)/2^4. \end{array}\right. } \end{aligned}$$

(3.88)

(Explicitly, $(7!!)/2^4 = 105/16$.)

Without going into further details we note that also the higher-order examples $n \in {{\mathbb {N}}}$, $n \ge 3$, can be explicitly solved in terms generalized hypergeometric functions and Meijer’s G-function (this is discussed in [14]).

4 On the Real Part of the Roots of $D_{2n}(\,\cdot \,;c)$, $c\in {{\mathbb {R}}}$

For $n\in {{\mathbb {N}}}$ and $c\in {{\mathbb {R}}}$, let $D_{2n}(\,\cdot \,;c)$ be the polynomial given by (3.22) and note that all of its coefficients are real. The goal of this section is to determine how many of the roots of $D_{2n}(\,\cdot \,;c)$ have real part $> - 1/2$. Results of this sort are typically approached by using the Routh–Hurwitz criterion. We propose a different approach here, even though Hurwitz’s ideas still play a central role.

Let us begin by fixing some notation. For $c\in {{\mathbb {R}}}$, let the roots of $D_{2n}(\,\cdot \,;c)=0$ be denoted $\alpha _j(c)$, $j=1,\ldots ,2n$. By the continuous dependence of the roots of a polynomial on the coefficients (see [33, Theorem (1.4)]), we may choose our labeling such that each $\alpha _j(c)$ is a continuous function of c and

$$\begin{aligned} {\text {Re}}(\alpha _1(c)) \le {\text {Re}}(\alpha _2(c)) \le \cdots \le {\text {Re}}(\alpha _n(c)) \le \cdots \le {\text {Re}}(\alpha _{2n}(c)), \quad c\in {{\mathbb {R}}}.\nonumber \\ \end{aligned}$$

(4.1)

Note that ${\text {Re}}(\alpha _j(0)) =\alpha _j(0) =j-1$ for $j=1,\ldots ,2n$. The fact that

$$\begin{aligned} D_{2n}(\,\cdot \,;0) \text{ has } \text{2n } \text{ distinct } \text{ real } \text{ roots } >- 1/2 \end{aligned}$$

(4.2)

will be of crucial importance in all that follows.

Example 4.1

Figure 1 shows the graphs of the real parts of the roots of $D_6(\,\cdot \,;c)$ as functions of $c\in {\mathbb {R}}$. The scale for the x-axis has been chosen such that $x=c^{1/6}$ for $c>0$ and $x={\text {sgn}}(c)|c|^{1/6}$ for $c<0$. The dotted lines show the graphs of the real parts of the roots of $(\,\cdot \,)^6 - c=0$ as functions of c. One notes that these dotted lines are straight lines precisely because of our special choice of scale for the x-axis. Furthermore, as $c\rightarrow \pm \infty$, the graph of each function ${\text {Re}}(\alpha _j(c))$ approaches one of these straight lines asymptotically. One observes that for $c\ll 0$, one has ${\text {Re}}(\alpha _1(c))={\text {Re}}(\alpha _2(c))< {\text {Re}}(\alpha _3(c))={\text {Re}}(\alpha _4(c))< {\text {Re}}(\alpha _5(c))={\text {Re}}(\alpha _6(c))$. Similarly, for $c\gg 0$, one infers that ${\text {Re}}(\alpha _1(c))<{\text {Re}}(\alpha _2(c))= {\text {Re}}(\alpha _3(c))<{\text {Re}}(\alpha _4(c))= {\text {Re}}(\alpha _5(c))<{\text {Re}}(\alpha _6(c))$.

As will be shown later, we have

$$\begin{aligned} \begin{aligned} {\text {Re}}(\alpha _1(c)) \le -\frac{1}{2}&\quad \text{ iff } \quad c\le \frac{2240 \left( 214-7 \sqrt{1009}\right) }{27} \approx -693.0 \\&\qquad \quad \text{ or } \quad c\ge \frac{10395}{64}\approx 162.4, \\ {\text {Re}}(\alpha _2(c)) \le -\frac{1}{2}&\quad \text{ iff } \quad c\le \frac{2240 \left( 214-7 \sqrt{1009}\right) }{27} \approx -693.0 \\&\qquad \quad \text{ or } \quad c\ge \frac{2240 \left( 214+7 \sqrt{1009}\right) }{27}\approx 36201.2, \\ {\text {Re}}(\alpha _3(c)) \le -\frac{1}{2}&\quad \text{ iff } \quad c\ge \frac{2240 \left( 214+7 \sqrt{1009}\right) }{27}\approx 36201.2, \end{aligned} \end{aligned}$$

(4.3)

where the algebraic numbers on the right are roots of the quadratic equation $27c^2 - 958720 c - 677376000 = 0$. If $j\in \{4,5,6\}$, then ${\text {Re}}(\alpha _j(c)) > - 1/2$ for all $c\in {{\mathbb {R}}}$.

Fig. 1

Graphs of the real parts of the roots of $D_6(\,\cdot \,;c)$ as functions of $c\in {\mathbb {R}}$

The proof of our main result, Theorem 4.5, concerning the real parts of the roots of $D_{2n}(\,\cdot \,;c)$, $c\in {\mathbb {R}}$, will depend on three lemmas. The first lemma states that for any $c\in {\mathbb {R}}$, the polynomial $D_{2n}(\,\cdot \,;c)$ cannot have more than two roots (counting multiplicity) having the same real part. More precisely, we have the following result:

Lemma 4.2

For $j,j'\in \{1,2,\ldots ,2n\}$ and $c\in {{\mathbb {R}}}$,

$$\begin{aligned} {\text {Re}}(\alpha _j(c)) ={\text {Re}}(\alpha _{j'}(c)) \, \text { implies } \, |j-j'|\le 1, \end{aligned}$$

(4.4)

Furthermore, if ${\text {Re}}(\alpha _j(c)) ={\text {Re}}(\alpha _{j'}(c))$ and $|j-j'|=1$, then $\alpha _j(c),\alpha _{j'}(c)\not \in {\mathbb {R}}$ and $\overline{\alpha _j(c)}= \alpha _{j'}(c)$.

Proof

Let $c\in {\mathbb {R}}$ and note that

$$\begin{aligned} \frac{d}{dz} D_{2n}(z;c) = \frac{d}{dz} \left( D_{2n}(z;0)+(-1)^nc\right) = \frac{d}{dz} D_{2n}(z;0), \quad z\in {\mathbb {C}}. \end{aligned}$$

(4.5)

In particular, $D_{2n}(\,\cdot \,;c)$ and $D_{2n}(\,\cdot \,;0)$ have the same critical points. By (4.2)

$$\begin{aligned} \text{ all } \text{ of } \text{ the } \text{ roots } \text{ of } \text{ the } \text{ derivative } \text{ of } D_{2n}(\,\cdot \,;0) \text{ are } \text{ real } \text{ and } \text{ simple, } \end{aligned}$$

(4.6)

and hence it follows that $D_{2n}(\,\cdot \,;c)$ does not have real roots of multiplicity greater than two. Moreover, since $c\in {\mathbb {R}}$, all roots of $D_{2n}(\,\cdot \,;c)$ are real or complex conjugates. Arguing by contradiction, suppose the polynomial $D_{2n}(\,\cdot \,;c)$ has more than two roots (counting multiplicity) having the same real part. Then

$$\begin{aligned} \begin{aligned}&\text{ there } \text{ exist } \text{ two } \text{ roots } z_1,z_2\in {{\mathbb {C}}}\text { of }D_{2n}(\,\cdot \,;c)\text { such that}\\&\quad {\text {Re}}(z_1) ={\text {Re}}(z_2)\text { and }0 \le {\text {Im}}(z_1)<{\text {Im}}(z_2). \end{aligned} \end{aligned}$$

(4.7)

We now use the Grace–Heawood theorem to obtain a contradiction. More precisely, we use the following corollary of (the proof of) the Grace–Heawood theorem, which is stated on page 126 of [38] as a “Supplement”:

If $z_1,z_2\in {{\mathbb {C}}}$ are two distinct roots of a complex polynomial of degree $\ge 2$, then neither of the two closed half-planes whose boundary is the perpendicular bisector of the line segment $[z_1,z_2]$ is devoid of any critical points of the polynomial.

When applied to the two roots $z_1,z_2$ of $D_{2n}(\,\cdot \,;c)$ as in the claim, this leads to a contradiction as follows. Note that the perpendicular bisector of the line segment $[z_1,z_2]$ in our situation is of the form $\{z\in {{\mathbb {C}}}\mid {\text {Im}}(z) = y_0\}$, where $y_0:=[{\text {Im}}(z_1)+{\text {Im}}(z_2)]/2>0$. Now recall that by (4.6) all the critical points of $D_{2n}(\,\cdot \,;c)$ are real. Thus, the closed half-plane $\{z\in {{\mathbb {C}}}\,| {\text {Im}}(z)\ge y_0 \}$ would be devoid of any critical points of $D_{2n}(\,\cdot \,;c)$. This is the desired contradiction.

$\square$

The second lemma is concerned with the asymptotic behavior of the real parts of the roots of $D_{2n}(\,\cdot \,;c)$ as $c\rightarrow \pm \infty$.

Lemma 4.3

For $j\in \{1,2,\ldots ,2n\}$ and $c\in {{\mathbb {R}}}$,

$$\begin{aligned} \lim _{c\rightarrow + \infty } {\text {Re}}(\alpha _j(c)) = {\left\{ \begin{array}{ll} -\infty , &{} 1\le j \le n, \\ + \infty , &{} n+1\le j \le 2n, \end{array}\right. } \end{aligned}$$

(4.8)

and

$$\begin{aligned} \lim _{c\rightarrow -\infty } {\text {Re}}(\alpha _j(c)) = {\left\{ \begin{array}{ll} -\infty , &{} 1\le j \le n-1, \\ \textstyle {n-(1/2)}, &{} n \le j\le n+1, \\ + \infty , &{}n+2\le j \le 2n. \end{array}\right. } \end{aligned}$$

(4.9)

Proof

For the purpose of this proof, let $f(\cdot )$ be the polynomial given by

$$\begin{aligned} f(z):=D_{2n}(z+(n-(1/2));0),\quad z\in {{\mathbb {C}}}. \end{aligned}$$

(4.10)

The half-integer $n-(1/2)$ is the center of mass of the roots of $D_{2n}(\,\cdot \,;0)$ and hence the center of mass of the roots of $f(\,\cdot \,)$ is 0. In other words,

$$\begin{aligned} \text{ if } \text{ we } \text{ write } f(z)=\sum _{j=0}^{2n}a_j z^j\text{, } \text{ then } a_{2n-1}=0. \end{aligned}$$

(4.11)

For $z_0\in {{\mathbb {C}}}$, it will be convenient to define polynomials $f(\,\cdot \,;z_0)$ and $g(\,\cdot \,;z_0)$ by

$$\begin{aligned} f(z;z_0):=f(z)-z_0^{2n}, \quad g(z;z_0):=z^{2n}-z_0^{2n}, \quad z\in {{\mathbb {C}}}. \end{aligned}$$

(4.12)

One notes that if $z_0^{2n}=(-1)^{n-1}c$, then $f(z;z_0)=D_{2n}(z-(1/2);c)$ for all $z\in {{\mathbb {C}}}$.

Next, let $\varepsilon >0$. We claim that there exists a real number $R >0$ such that if $|z_0|>r$, then the polynomial $f(\,\cdot \, ;z_0)$ has a unique root in the open disc $U(z_0;\varepsilon ):=\{z\in {\mathbb {C}} \,|\, |z-z_0|<\varepsilon \}$. Notice that $g(\,\cdot \, ;z_0)$ has a unique root in $U(z_0;\varepsilon )$, namely $z_0$, as long as $|z_0|$ is sufficiently large. Thus, one can use Rouché’s theorem as follows. Let $M:=\max \{|a_{2n-2}|,\ldots , |a_1|,|a_{0}|\}$. If $|z_0|\ge 1+\varepsilon$ and $z\in \partial U(z_0;\varepsilon )$, then $1\le |z|\le |z_0|+\varepsilon$ and hence (keeping in mind (4.11))

$$\begin{aligned} \begin{aligned} |f(z;z_0)-g(z;z_0)|&=|a_{2n-2}z^{n-2}+\ldots +a_1z+a_0|\\&\le |a_{2n-2}| |z|^{2n-2}+ \cdots + |a_1| |z|+|a_0| \\&\le M(|z|^{2n-2}+\cdots + |z|+1) \\&\le (2n-1)M |z|^{2n-2}\\&\le (2n-1)M(|z_0|+\varepsilon )^{2n-2}. \end{aligned} \end{aligned}$$

(4.13)

Furthermore, if $|z_0|\ge 1+ \varepsilon$, then the minimum of $|g(\,\cdot \, ;z_0)|$ on the boundary $\partial U(z_0;\varepsilon )$ is attained at $z=(|z_0|- \varepsilon ) z_0/|z_0|$ and hence for every $z\in \partial U(z_0;\varepsilon )$ one has

$$\begin{aligned} \begin{aligned} |g(z;z_0)|&=|z^{2n}-z_0^{2n}|\ge |(|z_0|-\varepsilon )^{2n}-|z_0|^{2n}|\\&=\varepsilon \, |(|z_0|-\varepsilon )^{2n-1}+\cdots + (|z_0|-\varepsilon )+1|. \end{aligned} \end{aligned}$$

(4.14)

One notes that if $|z_0|$ is sufficiently large, then

$$\begin{aligned} \varepsilon \, [(|z_0|-\varepsilon )^{2n-1}+\cdots + (|z_0|-\varepsilon )+1] > (2n-1)M(|z_0|+\varepsilon )^{2n-2} \end{aligned}$$

(4.15)

since the left-hand side is a polynomial in $|z_0|$ of degree $2n-1$ (with positive leading coefficient) and the right-hand side is a polynomial in $|z_0|$ of degree $2n-2$ (with positive leading coefficient.) Therefore, if $|z_0|$ is sufficiently large, then

$$\begin{aligned} |g(z;z_0)| > |f(z;z_0)-g(z;z_0)|\quad \text{ for } \text{ every } z\in \partial U(z_0;\varepsilon ) \end{aligned}$$

(4.16)

and hence, by Rouché’s theorem, $f(\,\cdot \,;z_0)$ and $g(\,\cdot \,;z_0)$ have the same number of roots (counted with multiplicity) in $U(z_0;\varepsilon )$. It follows that there exists some $R>0$ such that if $|z_0|>R$, then $f(\,\cdot \,;z_0)$ has a unique root in the open disc $U(z_0;\varepsilon )$.

We can now complete the proof of Lemma 4.3. For $c\in {{\mathbb {R}}}$, let the roots of

$$\begin{aligned}{}[z-(n-(1/2))]^{2n}+(-1)^nc=0, \quad z\in {{\mathbb {C}}}, \end{aligned}$$

(4.17)

be denoted $\beta _j(c)$, $j=1,\ldots ,2n$. One can choose a labeling such that

$$\begin{aligned} {\text {Re}}(\beta _1(c)) \le {\text {Re}}(\beta _2(c)) \le \cdots \le {\text {Re}}(\beta _n(c)) \le \cdots \le {\text {Re}}(\beta _{2n}(c)), \quad c\in {{\mathbb {R}}}.\qquad \quad \end{aligned}$$

(4.18)

There is a statement analogous to Lemma 4.2 for the roots $\beta _{j}(c)$, $j=1,\ldots ,2n$. In light of this, there is a “canonical” labeling for both the roots $\alpha _{j}(c)$ and $\beta _{j}(c)$ such that if $1\le j<2n$ and ${\text {Re}}(\alpha _j(c))= {\text {Re}}(\alpha _{j+1}(c))$   [resp. ${\text {Re}}(\beta _j(c))= {\text {Re}}(\beta _{j+1}(c))$], then ${\text {Im}}(\alpha _j(c))<{\text {Im}}(\alpha _{j+1}(c))$   [resp. ${\text {Im}}(\beta _j(c))<{\text {Im}}(\beta _{j+1}(c))$]. The roots of (4.17) are trivial to determine and a straightforward (but somewhat tedious) analysis shows that the asymptotic behavior of ${\text {Re}}(\beta _j(c))$ as $c\rightarrow \pm \infty$ is given by (4.8) and (4.9), respectively, with $\alpha _j{(c)}$ replaced by $\beta _j{(c)}$,    $j=1,2\ldots , 2n$.

Now for $\varepsilon >0$ and $|c|\gg 0$, by the Rouché argument from above applied to $z_0=\beta _j(c)$,

$$\begin{aligned} |\alpha _{j}(c) - \beta _{j}(c)|<\varepsilon , \quad j=1,2\ldots , 2n. \end{aligned}$$

(4.19)

Therefore, the asymptotic behavior of ${\text {Re}}(\beta _j(c))$ as $c\rightarrow \pm \infty$ is given by (4.8) and (4.9), respectively. $\square$

Finally, the last lemma is related to the Routh–Hurwitz criterion, adapted to our situation. This takes some preparation. For $c\in {\mathbb {R}}$, one first expands $D_{2n}(z-(1/2);c)$ as a polynomial in z,

$$\begin{aligned} D_{2n}( z -(1/2);c)=q_{2n} z^{2n}+ q_{2n-1} z^{2n-1} + \cdots + q_1 z+ \big [q_0+(-1)^n c\big ],\qquad \quad \end{aligned}$$

(4.20)

and then considers the associated $(2n \times 2n)$ Hurwitz matrix,

$$\begin{aligned} H_{2n}(c):= \begin{pmatrix} q_{2n-1}\quad \!&{} \! q_{2n-3}\quad \! &{}\! q_{2n-5}\!\quad &{}\ \ \cdots &{} \ \ 0\ \ {} &{}\ \ 0 \ \ {} &{} \ \ 0 \ \ {} &{}\\ q_{2n}\quad &{} q_{2n-2}\quad &{} q_{2n-4}\quad &{}\ \ \ddots &{} \vdots &{} \vdots &{} \vdots &{}\\ 0\quad &{} q_{2n-1}\quad &{} q_{2n-3}\quad &{}\ \ \ddots &{} \vdots &{} \vdots &{} \vdots &{}\\ \vdots &{} q_{2n} &{} q_{2n-2}&{} &{} 0 &{}\vdots &{}\vdots &{}\\ \vdots &{} 0&{} q_{2n-1}\quad &{} &{} q_0\!+\!(-1)^n c &{}\vdots &{} \vdots &{}\\ \vdots &{} \vdots &{} q_{2n}\quad &{} &{} q_1 &{} 0 &{} \vdots &{}\\ \vdots &{} \vdots &{}0&{} &{} q_2 &{} q_0\! +\! (-1)^n c &{} \vdots &{}\\ \vdots &{} \vdots &{} \vdots &{} &{} q_3 &{} q_1 &{}0 &{} \\ 0 &{} 0 &{} 0 &{}\ \ \cdots &{} q_4 &{} q_2 &{} q_0\!+\!(-1)^n c &{}\\ \end{pmatrix}.\nonumber \\ \end{aligned}$$

(4.21)

One notes that $q_j\in {\mathbb {Q}}$ for all $j\in \{0,1,\ldots , 2n\}$. Furthermore, one observes that c only occurs in the even rows. This implies that the function $\det \left( H_{2n}(\,\cdot \,)\right)$ is a polynomial of degree n with rational coefficients. By Laplace expansion along the last column,

$$\begin{aligned} \det \left( H_{2n}(c)\right) = \left[ q_0+(-1)^n c\right] h_{n-1}(c), \end{aligned}$$

(4.22)

where $h_{n-1}(\,\cdot \,)$ is a polynomial of degree $n-1$ with rational coefficients. There is a simple closed expression for $q_0$, which is reminiscent of the expression on the right-hand side of (1.9):

$$\begin{aligned} q_0=\frac{(4n-1)!!}{2^{2n}}. \end{aligned}$$

(4.23)

Formula (4.23) is easily proved by induction using that

$$\begin{aligned} q_0=D_{2n}(-1/2;0)=\prod _{j=1}^{2n} [j-(1/2)]. \end{aligned}$$

(4.24)

Lemma 4.4

For $j\in \{1,2,\ldots ,2n\}$ and $c\in {{\mathbb {R}}}$, if ${\text {Re}}(\alpha _{j}(c))=- 1/2$, then

$$\begin{aligned} \det (H_{2n}(c))=0, \end{aligned}$$

(4.25)

that is,

$$\begin{aligned} c=(-1)^{n-1} q_0, \, \text { or, } \, h_{n-1}(c)=0, \end{aligned}$$

(4.26)

where $h_{n-1}(\,\cdot \,)$ is given by (4.22).

Proof

Note that the roots of the polynomial (4.20) are just the roots of $D_{2n}(\,\cdot \,; c)$ shifted by 1/2, that is, roots of the polynomial (4.20) are $\alpha _{j}(c)+(1/2)$, where $j\in \{1,2,\ldots ,2n\}$. It then follows from Orlando’s formula (see [13, § XV.7]) that

$$\begin{aligned} h_{n-1}(c)= \prod _{1\le j_1< j_2\le 2n} \{[\alpha _{j_1}(c) +(1/2)]+[\alpha _{j_2}(c) +(1/2)]\}. \end{aligned}$$

(4.27)

Next, let $j\in \{1,2,\ldots ,2n\}$ and $c\in {{\mathbb {R}}}$ such that ${\text {Re}}(\alpha _{j}(c))=-1/2$. First suppose $\alpha _{j}(c)\in {{\mathbb {R}}}$. Then $\alpha _{j}(c)=-1/2$ and $D_{2n}(-1/2; c) = D_{2n}(-1/2; 0)+(-1)^nc=0$, which implies that $c=(-1)^{n-1}q_0$. Next, suppose that $\alpha _{j}(c)\not \in {{\mathbb {R}}}$. By Lemma 4.2, there exists some $j'\in \{1,2,\ldots ,2n\}$, $j\not =j'$, such that $\alpha _{j'}(c)=\overline{\alpha _j(c)}$. Then $[\alpha _{j}(c) + (1/2)] + [\alpha _{j'}(c) + (1/2)] = 0$ and hence $h_{n-1}(c)=0$ by (4.27). $\square$

We now have all the necessary ingredients to prove the main result of this section, Theorem 4.5. In this context we will use the floor and ceiling notation: One recalls that for $n\in {{\mathbb {N}}}$, $\lfloor n/2 \rfloor$ denotes the greatest integer less than or equal to n/2; similarly, $\lceil n/2\rceil$ denotes the least integer greater than or equal n/2. Thus, for $n\in {{\mathbb {N}}}$, one has

$$\begin{aligned} \lceil n/2\rceil = {\left\{ \begin{array}{ll} \lfloor n/2\rfloor + 1 =(n+1)/2 &{} \text{ if } n \text{ is } \text{ odd },\\ \lfloor n/2\rfloor =n/2 &{} \text{ if } n \text{ is } \text{ even }. \end{array}\right. } \end{aligned}$$

(4.28)

Recalling Remark 3.3 (i), one obtains for $c \in {{\mathbb {R}}}$, $d \in (0,\infty )$,

$$\begin{aligned} \#\left( \tau _{2n}(c)|_{(0,d)}\right){} & {} = \text{ the } \text{ number } \text{ of } j\in \{1,2,\ldots ,2n\} \text{ such } \text{ that } {\text {Re}}(\alpha _j(c))\nonumber \\{} & {} > - 1/2. \end{aligned}$$

(4.29)

Theorem 4.5

(i) For every $n\in {\mathbb {N}}$, $n\ge 2$, there exist n real constants

$$\begin{aligned} c_{n}^{(1)}<c_{n}^{(2)}< \cdots < c_{n}^{(n)} \end{aligned}$$

(4.30)

such that the following items (a)–(c) hold:

(a):

For $c\in {\mathbb {R}}$, $d \in (0,\infty )$, one has

$$\begin{aligned} \#\left( \tau _{2n}(c)|_{(0,d)}\right) = {\left\{ \begin{array}{ll} n,&{} \text{ if }\quad c \ge c_{n}^{(n)},\\ n+2(n-k),&{} \text{ if }\quad c_{n}^{(k)}\le c<c_{n}^{(k+1)} \ \text{ and }\ \lfloor n/2 \rfloor< k\le n-1,\\ 2n, &{} \text{ if }\quad c_{n}^{(k)}< c<c_{n}^{(k+1)} \ \text{ and }\ k=\lfloor n/2 \rfloor ,\\ n+2k+1,&{} \text{ if }\quad c_{n}^{(k)}< c \le c_{n}^{(k+1)} \ \text{ and }\ 1\le k< \lfloor n/2 \rfloor ,\\ n+1,&{} \text{ if }\quad c\le c_{n}^{(1)}.\\ \end{array}\right. } \end{aligned}$$

(4.31)

(b):

The constant $c_{n}^{(\lceil n/2\rceil )}$ is given by the formula

$$\begin{aligned} c_{n}^{(\lceil n/2\rceil )} = (-1)^{n-1} \frac{(4n-1)!!}{2^{2n}}. \end{aligned}$$

(4.32)

(c):

The constants $c_{n}^{(1)},c_{n}^{(2)},\ldots c_{n}^{(\lceil n/2 \rceil -1)},c_{n}^{(\lceil n/2 \rceil +1)}, \ldots , c_{n}^{(n)}$ are the roots of the polynomial $h_{n-1}(\,\cdot \,)$ of degree $n-1$ with rational coefficients. In addition,

$$\begin{aligned} c_n^{(n)} \ge \frac{(4n-1)!!}{2^{2n}} \underset{n \rightarrow \infty }{=} 2^{1/2} (2/e)^n n^{2n}[1 + O(1/n)]. \end{aligned}$$

(4.33)

(ii) For $n=1$ one obtains

$$\begin{aligned} \#\left( \tau _{2}(c)|_{(0,d)}\right) = {\left\{ \begin{array}{ll} 1, &{} \text{ if }\quad c \ge 3/4, \\ 2, &{} \text{ if }\quad c < 3/4. \end{array}\right. } \end{aligned}$$

(4.34)

Proof

(i) The constants $c_{n}^{(1)},\ldots , c_{n}^{(n)}$ will turn out to be the roots of the polynomial $\det (H_{2n}(\,\cdot \,))$ of degree n given by (4.21). However, it is not clear, a priori, that $\det (H_{2n}(\,\cdot \,))$ has n distinct real roots. For that reason, we will have to define our constants differently.

Next, we recall that the polynomial $D_{2n}(\,\cdot \,;0)$ has 2n distinct real roots, namely the nonnegative integers $\alpha _{j}(0)=j-1$, where $j\in \{1,2\ldots , 2n\}$. In particular, ${\text {Re}}(\alpha _{j}(0))>- 1/2$ for all $j\in \{1,2\ldots , 2n\}$. By Lemma 4.3, if $1\le j\le n-1$, one has $\lim _{c\rightarrow -\infty }{\text {Re}}(\alpha _{j}(c))=-\infty$ and hence $\{c < 0 \,| {\text {Re}}(\alpha _{j}(c))=- 1/2\}$ is nonempty by continuity; similarly, if $1\le j\le n$, then $\lim _{c\rightarrow \infty }{\text {Re}}(\alpha _{j}(c))=-\infty$ and hence $\{c > 0 \,| {\text {Re}}(\alpha _{j}(c))=- 1/2\}$ is nonempty by continuity. Now, for $1\le k\le n$, define

$$\begin{aligned} c_{n}^{(k)}:= {\left\{ \begin{array}{ll} \min \{c \in {{\mathbb {R}}}\,| {\text {Re}}(\alpha _{n - 2k+1}(c))=- 1/2\} &{} \text{ if } 1\le k\le \lfloor n/2\rfloor \\ \max \{c \in {{\mathbb {R}}}\,| {\text {Re}}(\alpha _{2 (k - \lfloor n/2\rfloor ) - 1 }(c))=- 1/2\} &{} \text{ if } \lfloor n/2\rfloor < k\le n. \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.35)

One notes that if $1\le k\le \lfloor n/2\rfloor$, then $1\le n - 2k+1 \le n-1$ and $c_n^{(k)}<0$; similarly, if $\lfloor n/2\rfloor < k\le n$, then $1\le 2 (k - \lfloor n/2\rfloor ) - 1\le n$ and $c_n^{(k)}>0$. By (4.1), we then obtain

$$\begin{aligned} c_{n}^{(1)}\le c_{n}^{(2)}\le \cdots \le c_{n}^{(\lfloor n/2\rfloor )}< 0< c_{n}^{(\lfloor n/2\rfloor +1)}\le \cdots \le c_{n}^{(n-1)} \le c_{n}^{(n)}\nonumber \\ \end{aligned}$$

(4.36)

Next, we use Lemma 4.2 to show that all the inequalities in (4.36) are strict. Suppose $c_n^{(k)}=c_n^{(k+1)}$ for some $1\le k \le \lfloor n/2\rfloor -1$. Then ${\text{ Re }}\big (\alpha _{n-2k+1}(c_n^{(k)})\big )={\text{ Re }}\big (\alpha _{n-2k-1}(c_n^{(k)})\big )$ and since $|(n-2k+1)-(n-2k-1)|=2>1$, this contradicts (4.4). The same argument also yields a contradiction if $c_n^{(k)}=c_n^{(k+1)}$ for some $\lfloor n/2\rfloor < k \le n-1$. Therefore, all the inequalities in (4.36) are strict.

We can say a bit more about the constants $c_{n}^{(\lfloor n/2\rfloor )}$ and $c_{n}^{(\lfloor n/2\rfloor +1)}$.

Claim 4.6

We have

$$\begin{aligned} c_{n}^{(\lfloor n/2\rfloor )} \le -q_0< 0< q_0 \le c_{n}^{(\lfloor n/2\rfloor +1)}, \end{aligned}$$

(4.37)

where $q_0=D_{2n}(-1/2;0)$ as in (4.24).

By (4.36) and the discussion leading up to it, the claim follows if we show that for $c\in {{\mathbb {R}}}$, the polynomial $D_{2n}(\,\cdot \,; c)$ has no roots with real part equal to $-1/2$ if $|c|<q_0$. To prove the latter, we will use a simple argument due to Tallis and Gordon [47, Theorem 1(a)]. Consider the polynomial $f(\,\cdot \,)$ given by $f(z):=D_{2n}(z-(1/2);0)$, $z\in {{\mathbb {C}}}$. By (3.22),

$$\begin{aligned} f(z)=\prod _{j=1}^{2n}\left[ (z-(1/2))-(j-1)\right] =\prod _{j=1}^{2n}\left[ z-(j-(1/2))\right] ,\quad z\in {{\mathbb {C}}}.\nonumber \\ \end{aligned}$$

(4.38)

Note that $D_{2n}(\,\cdot \,; c)$ has a root with real part equal $-1/2$ if and only if $f(\,\cdot \,)+(-1)^n c$ has a root on the imaginary axis. Suppose $f(ib)+(-1)^nc=0$ for some $b\in {{\mathbb {R}}}$. Then

$$\begin{aligned} |c|=|f(ib)|=\prod _{j=1}^{2n}|ib-(j-(1/2))|\ge \prod _{j=1}^{2n}[j-(1/2)]=q_0, \end{aligned}$$

(4.39)

which proves Claim 4.6.

Combining (4.36) and (4.37) implies

$$\begin{aligned} c_n^{(n)} \ge q_0 = \frac{(4n-1)!!}{2^{2n}} = \frac{\Gamma (4n)}{2^{4n-1} \Gamma (2n)}. \end{aligned}$$

(4.40)

Stirling’s formula (see, e.g., [1, No. 6.1.37]),

$$\begin{aligned} \Gamma (z) \underset{\begin{array}{c} z \rightarrow \infty \\ |\arg (z)|< \pi \end{array}}{=} (2 \pi )^{1/2} e^{-z} z^{z - (1/2)} [1 + O(1/z)], \end{aligned}$$

(4.41)

then yields (4.33).

By Lemma 4.4, $\det \big (H_{2n}\big (c_n^{(k)}\big )\big )=0$ for every $1\le k\le n$. Since the constants $c_n^{(k)}$ are distinct and since $\det (H_{2n}(\,\cdot \,))$ is a polynomial of degree n, the polynomial $\det (H_{2n}(\,\cdot \,))$ does not have any other roots. Furthermore, one of the constants $c_n^{(k)}$ must be equal to $(-1)^{n-1}q_0$ and the other $n-1$ constants must be the roots of the polynomial $h_{n-1}(\,\cdot \,)$, see (4.22). If n is odd, then $(-1)^{n-1}q_0=q_0>0$ and it follows from (4.37) and (4.36) that $c_{n}^{(\lfloor n/2\rfloor +1)}=q_0$; similarly, if n is even, then $(-1)^{n-1}q_0=-q_0<0$ and it follows from (4.37) and (4.36) that $c_{n}^{(\lfloor n/2\rfloor )}=-q_0$. In either case, in light of (4.28), we have $c_{n}^{(\lceil n/2\rceil )} =(-1)^{n-1}q_0$. Thus, recalling the formula for $q_0$ from (4.23), we obtain (4.32). This completes the proof of parts (b) and (c) of Theorem 4.5.

Before we prove part (a), we recall that by the continuity argument given in the first paragraph of this proof, for every $1\le j\le n-1$, there exists some $c<0$ such that ${\text {Re}}(\alpha _j(c))=-1/2$. By our observations above, this c must be one of the constants $c_{n}^{(k)}$ with $1\le k\le \lfloor n/2\rfloor$. Similarly, for every $1\le j\le n$, there exists some $c>0$ such that ${\text {Re}}(\alpha _j(c))=-1/2$ and, by our observations above, this c must be one of the constants $c_{n}^{(k)}$ with $\lfloor n/2\rfloor +1 \le k\le n$.

We will now prove part (a) in the case when n is odd. Then $n-1$ is even and $n-1=2\lfloor n/2\rfloor =2(n-\lceil n/2 \rceil )$. By Lemma 4.2 and since $n-1=2\lfloor n/2\rfloor$, for every $1\le k\le \lfloor n/2\rfloor$, there are exactly two distinct $j,j'\in \{1,2\ldots , n-1\}$ such that ${\text {Re}}\big (\alpha _j\big (c_{n}^{(k)}\big )\big )={\text {Re}}\big (\alpha _{j'}\big (c_{n}^{(k)}\big )\big )$. Furthermore, $c_{n}^{(\lfloor n/2\rfloor +1)}= c_{n}^{(\lceil n/2\rceil )}=q_0$ and $\alpha _{1}\big (c_{n}^{(\lceil n/2\rceil )}\big )=- 1/2\in {{\mathbb {R}}}$. By Lemma 4.2 and since $n-1=2(n-\lceil n/2 \rceil )$, for every $\lceil n/2\rceil +1\le k\le n$, there are exactly two distinct $j,j'\in \{2,3\ldots , n\}$ such that ${\text {Re}}\big (\alpha _j\big (c_{n}^{(k)}\big )\big )={\text {Re}}\big (\alpha _{j'}\big (c_{n}^{(k)}\big )\big )$. The resulting situation is summarized in Fig. 2a. We now use Fig. 2a to understand how the value of $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ changes with $c\in {{\mathbb {R}}}$. For $c\le c_{n}^{(1)}$, Fig. 2b shows that ${\text {Re}}(\alpha _{j}(c)) > - 1/2$ if and only if $n\le j \le 2n$. Therefore, $\#\left( \tau _{2n}(c)|_{(0,d)}\right) =n+1$ for $c\le c_{n}^{(1)}$. As c increases beyond $c_{n}^{(1)}$, the value of $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ jumps from $n+1$ to $n+3$ since for $c_{n}^{(1)}<c\le c_{n}^{(2)}$, ${\text {Re}}(\alpha _{j}(c)) > - 1/2$ if and only if $n-2\le j \le 2n$ (assuming that $n\ge 3$). As c increases more, the value of $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ increases by 2 each time c crosses one of the constants $c_{n}^{(k)}$ until c reaches $c_{n}^{(\lfloor n/2 \rfloor )})$, when the value $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ only increases by 1 from $2n-1$ to 2n. From then on, the value of $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ starts decreasing by 2 each time c moves beyond one of the constants $c_{n}^{(k)}$ until, finally, c passes $c_{n}^{(n)}$, and we have $\#\left( \tau _{2n}(c)|_{(0,d)}\right) =n$ since for $c\ge c_{n}^{(n)}$, ${\text {Re}}(\alpha _{j}(c)) > - 1/2$ if and only if $n+1\le j \le 2n$. The result is the piecewise-formula for $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ stated in part (a).

In the case when n is even, the argument is, mutatis mutandis, the same. The situation is summarized in Fig. 2b. The result is the same piecewise-formula for $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ stated in part (a).

(ii) This has been discussed in Example 3.5. $\square$

Fig. 2

The constants $c_{n}^{(k)}$

Corollary 4.7

For every $n\in {\mathbb {N}}$, there exists a positive constant $c_n\in {{\mathbb {R}}}$ such that

$$\begin{aligned} \big \{c\in {\mathbb {R}} \,\big |\, \#\left( \tau _{2n}(c)|_{(0,d)}\right) =n \big \}=[c_n,\infty ), \end{aligned}$$

(4.42)

and thus,

$$\begin{aligned} \begin{aligned}&T_{2n, min}(c) \text{ is } \text{ self-adjoint } \big (\text{ equivalently, } {\overset{{{\textbf {..}}}}{T}}_{2n, min} \text{ is } \text{ essentially } \text{ self-adjoint }\big ) \\ {}&\quad \text{ in } L^2((0,\infty );dx) \text{ if } \text{ and } \text{ only } \text{ if } \ c \ge c_n. \end{aligned} \end{aligned}$$

(4.43)

In addition,

$$\begin{aligned} c_1 = 3/4, \quad c_n = c_n^{(n)} \ge \frac{(4n-1)!!}{2^{2n}}, \; n \in {{\mathbb {N}}}, \; n \ge 2 \end{aligned}$$

(4.44)

(see (4.30), (4.31), and (4.40)).

Put differently, Corollary 4.7 asserts there exist no “islands” (i.e., intervals or its degeneration to points) of nonessential self-adjointness for $\tau _{2n}(c)$ $\big |_{C_0^{\infty }((0,\infty ))}$ for $c \ge c_n$.

We explicitly record the following exact expressions:

$$\begin{aligned} c_{1}&= 3/4, \nonumber \\ c_{2 }&= 45, \nonumber \\ c_{3 }&= 2240 \big (214+7 \sqrt{1009}\big )\big /27 \approx 36201.1645283357, \nonumber \\ c_{4 }&=2835 \Bigg ( 13711+\frac{190309441}{\root 3 \of {2625188010911+1805760 \sqrt{-292868607}}} \nonumber \\&\quad + \root 3 \of {2625188010911+1805760 \sqrt{-292868607}}\ \Bigg ) \nonumber \\&=38870685+5670 \sqrt{\frac{292868607}{127}} \sin \left( \frac{1}{3} \tan ^{-1}\left( \frac{9 \sqrt{292868607}}{466120}\right) \right) \nonumber \\&\quad +\frac{876128400}{\sqrt{127}} {\cos \left( \frac{1}{3} \tan ^{-1}\left( \frac{9 \sqrt{292868607}}{466120}\right) \right) }\nonumber \\&\approx 117089256.9368802. \end{aligned}$$

(4.45)

Corollary 4.8

For every $n\in {\mathbb {N}}$ and every $m\in \{n,n+1,\cdots ,2n\}$, there exists some $c\in {\mathbb {R}}$ such that $\#\left( \tau _{2n}(c)|_{(0,d)}\right) =m$.

Proof

By Theorem 4.5, as c increases from $c\ll 0$ to $c\gg 0$, $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ takes on the values

$$\begin{aligned} n+1,n+3, \ldots ,2n-2, 2n, 2n-1, 2n-3, \ldots , n+2,n, \quad \text{ if } n \text{ is } \text{ odd, }\qquad \quad \end{aligned}$$

(4.46)

and

$$\begin{aligned} n+1,n+3, \ldots , 2n-3, 2n-1, 2n, 2n-2, \ldots , n+2,n, \quad \text{ if } n \text{ is } \text{ even }.\qquad \quad \end{aligned}$$

(4.47)

In either case, $\#\left( \tau _{2n}(c)|_{(0,d)}\right)$ takes on all integer values from n to 2n. $\square$

In particular, Corollary 4.8 proves that every possible integer in the interval [n, 2n] in (3.23) is attained for some $c \in {{\mathbb {R}}}$.

Example 4.9

If $n=3$, then $q_0=10395/64$ and

$$\begin{aligned} h_2(c)&=\left| \begin{array}{ccccc} 18 &{} 435 &{} {4881}/{8} &{} 0 &{} 0\\ -1 &{} -{505}/{4} &{} -{12139}/{16} &{} c-10395/64 &{} 0 \\ 0 &{} 18 &{} 435 &{} {4881}/{8} &{} 0 \\ 0 &{} -1 &{} -505/4 &{} -12139/16 &{} c-10395/64 \\ 0 &{} 0 &{} 18 &{} 435 &{} 4881/8 \\ \end{array} \right| \nonumber \\&= -5832 c^2+207083520 c+146313216000, \quad c \in {{\mathbb {R}}}. \end{aligned}$$

(4.48)

The roots of $h_2(\,\cdot \,)$ are $2240\left( 214\pm 7 \sqrt{1009}\right) /27$. Therefore, by Theorem 4.5 one finds

$$\begin{aligned} \#\left( \tau _{6}(c)|_{(0,d)}\right) = {\left\{ \begin{array}{ll} 3,&{} \text{ if }\quad 2240 \left( 214+7 \sqrt{1009}\right) /27\le c;\\ 5,&{} \text{ if }\quad 10395/64\le c< 2240\left( 214+7 \sqrt{1009}\right) /27;\\ 6,&{} \text{ if }\quad 2240\left( 214-7 \sqrt{1009}\right) /27< c<10395/64;\\ 4,&{} \text{ if }\quad c\le 2240\left( 214-7 \sqrt{1009}\right) /27. \end{array}\right. }\nonumber \\ \end{aligned}$$

(4.49)

Appendix A. Some Conjectures

In this section, when dealing with polynomials, we will view them as elements in a polynomial ring as in abstract algebra. We will review some standard notational conventions and basic results. Let X be an indeterminate (formal symbol). We denote by ${{\mathbb {Z}}}[X]$ (resp. ${{\mathbb {Q}}}[X]$) the ring of polynomials in the indeterminate X with coefficients in ${{\mathbb {Z}}}$ (resp. ${{\mathbb {Q}}}$). A polynomial $f(X)\in {{\mathbb {Q}}}[X]$ is called irreducible, if it has positive degree and it cannot be written as a product $f(X)=g(X)h(X)$, where $g(X),h(X)\in {{\mathbb {Q}}}[X]$ are polynomials of degree strictly less than the degree of f(X).

Conjecture A.1

For $n\in {\mathbb {N}}$, $n\ge 2$, the polynomial

$$\begin{aligned} g_{n-1}(X):= \frac{(-1)^{\lfloor n/2 \rfloor }}{{(2 n^2)}^{n}}\, h_{n-1}(X) \end{aligned}$$

(A.1)

is a monic irreducible polynomial in ${\mathbb {Q}}[X]$ of degree $n-1$ with Galois group $S_{n-1}$. In particular, for $n\ge 6$, the constants $c_{n}^{(1)},c_{n}^{(2)},\ldots c_{n}^{(\lceil n/2 \rceil -1},c_{n}^{(\lceil n/2 \rceil +1)},$ $\ldots , c_{n}^{(n)}$ are algebraic numbers that are not expressible in radicals over ${\mathbb {Q}}$.

Proof for $n=5$. We have

$$\begin{aligned} \begin{aligned} g_4(X)&= X^4-5237598744576 X^3/5-3477424021724410819117056X^2/3125\\&\quad +2933863158888223380395161288704X/125\\&\quad +246639641224100448713004224731938816/55. \end{aligned} \end{aligned}$$

(A.2)

Let ${\widetilde{g}}_4(X):=(3125)^4\, g_4(X/3125)$. Then ${\widetilde{g}}_4(X)$ is a monic polynomial of degree 4 with integer coefficients. Reducing the coefficient of modulo 19, one obtains

$$\begin{aligned} {\widetilde{g}}_4(X)\equiv X^4+11 X^3+3 X^2+11 X+15\ {\text {mod}} 19. \end{aligned}$$

(A.3)

It is easy to check that $X^4+11 X^3+3 X^2+11X+15$ is irreducible modulo 19. By Gauss’ lemma, it follows that $g_4(X)$ is irreducible over ${{\mathbb {Q}}}$. $\square$

Proof for $n=6$. We have

$$\begin{aligned} g_5(X)&=X^5-15354318108567042605X^4/729\nonumber \\&\quad -333441081709503846926848000000X^3/3\nonumber \\&\quad +4983404391409567436628431599042560000000 X^2\nonumber \\&\quad +8770826733513986444066497798757941248000000000000X\nonumber \\&\quad -2088913117666248881257824386993081779822264320000000000000\qquad \end{aligned}$$

(A.4)

Let ${\widetilde{g}}_5(X):=(729)^5\, g_5(X/729)$. Then ${\widetilde{g}}_5(X)$ is a monic polynomial of degree 5 with integer coefficients. Note that $g_5(X)$ is irreducible over ${\mathbb {Q}}$ if and only if ${\widetilde{g}}_5(X)$ is irreducible over ${\mathbb {Q}}$. Furthermore, the Galois group of $g_5(X)$ is isomorphic to the Galois group of ${\widetilde{g}}_5(X)$. To prove the irreducibility and to compute the Galois group, we reduce the coefficients of ${\widetilde{g}}_5(X)$ modulo the primes 23 and 109:

$$\begin{aligned} {\widetilde{g}}_5(X)&\equiv X^5+5X^4+11 X^3+7 X^2+13X+16\ {\text {mod}} 23, \end{aligned}$$

(A.5)

$$\begin{aligned} {\widetilde{g}}_5(X)&\equiv (X^2+38 X+24)(X+42)(X+41)(x+11)\ {\text {mod}} 109. \end{aligned}$$

(A.6)

It is easy to check that $X^5+5 X^4+11 X^3+7 X^2+13X+16$ is irreducible modulo 23. Therefore, the polynomial ${\widetilde{g}}_5(X)$ is irreducible over ${\mathbb {Z}}$ and also over ${\mathbb {Q}}$ by Gauss’ lemma. It also follows, by a theorem due to Dedekind (see [26, Thm. 4.37]), that the Galois group of the polynomial ${\widetilde{g}}_5(X)$ contains a 5-cycle. Since the reduction of ${\widetilde{g}}_5(X)$ modulo 109 is the product an irreducible quadratic polynomial and three linear polynomials, Dedekind’s theorem implies that the Galois group of ${\widetilde{g}}_5(X)$ contains a transposition (2-cycle). A subgroup of $S_5$ that contains a transposition and a 5-cycle is $S_5$. Since $S_5$ is not a solvable group, Galois’ theorem, implies that $g_5(X)$ is not solvable and hence $c_{6}$ cannot be written in terms of radicals. $\square$

The same idea can be used to prove the conjecture for larger n. The following table shows what primes are used to verify the conjecture for $4\le n\le 12$.

We conclude with a vexing open conjecture:

Conjecture A.2

One has $\big ($recalling $c_n = c_n^{(n)}$ $\big )$

$$\begin{aligned} c_{n} \underset{n \rightarrow \infty }{\sim }\ \big (2n^2\big /\pi \big )^{2n}. \end{aligned}$$

(A.7)

and refer to [15] for numerical evidence in this context.