Essential self-adjointness of even-order, strongly singular, homogeneous half-line differential operators

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 \[ \tau_{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}}, \quad x>0, \; n \in \mathbb{N}, \; c \in \mathbb{R}, \] 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 $c \geq 3/4$, the case $n \in \mathbb{N}$, $n \geq 2$, is far from obvious. In particular, it is not at all clear from the outset that \[ \text{ there exists } c_n \in \mathbb{R}, \, n \in \mathbb{N}, \text{ such that } \tau_{2n}(c)\big|_{C_0^\infty((0,\infty))} \, \text{ is essentially self-adjoint if and only if } c \geq c_n. \tag{*}\label{0.1} \] As one of the principal results of this paper we indeed establish the existence of $c_n$, satisfying $c_n \geq (4n-1)!!\big/2^{2n}$, such that property \eqref{0.1} holds. In sharp contrast to the analogous lower semiboundedness question, \[ \text{ for which values of } c \, \text{\it is } \tau_{2n}(c)\big|_{C_0^{\infty}((0,\infty))} \, \text{ bounded from below?}, \] which permits the sharp (and explicit) answer $c \geq [(2n -1)!!]^{2}\big/2^{2n}$, $n \in \mathbb{N}$, the answer for \eqref{0.1} is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. For completeness we record explicitly, \[ c_1 = 3/4, \quad c_2= 45, \quad c_3 = 2240 \big(214+7 \sqrt{1009}\,\big)\big/27, \] 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 $n \geq 6$).

In the special case n = 1 it is well-known that the precise answer is (see, e.g., [41]), (1.4) holds for n = 1 if and only if c ≥ c 1 = 3/4.
(1.5) A priori it is not clear at all that this extends to n ∈ N, n ≥ 2, that is, it is not obvious from the outset that there exists c n ∈ R, n ∈ N, such that τ 2n (c) C ∞ 0 ((0,∞)) is essentially self-adjoint if and only if c ≥ c n . (1.6) Our principal new results, Theorem 4.5 and Corollary 4.6 assert that (1.6) indeed holds for some c n ∈ R satisfying c n ≥ (4n − 1)!! 2 2n , n ∈ N. (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 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 Q; we conjecture that this actually continues to hold for general n ≥ 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: T 2n,min (c) is bounded from below, and then actually, T 2n,min (c) ≥ 0, n ∈ N, if and only if c ≥ − [(2n − 1)!!] 2 2 2n . (1.9) This is a consequence of the sequence of sharp Birman-Hardy-Rellich inequalities, see Birman [5, p. 46] (see also Glazman [15,) (1.10) For more details on (1.10) see [14] and the extensive literature cited therein.
To decide the limit point property of τ 2n (c) C ∞ 0 ((0,∞)) at x = 0, one next argues that it suffices to choose µ = 0 in (1.11) which then leads to a special Euler-type equations which generically has solutions of power-type y j (0, x; c) = C j x α j (c) , 1 ≤ j ≤ 2n, (1.12) with α j (c), 1 ≤ j ≤ 2n, being the solutions of the underlying discriminant or indicial equation, In exceptional cases, where some of the α k (c) coincide, (1.12) is replaced by y k (0, x; c) = C k x α k (c) P (ln(x)), (1.14) where P ( • ) is a polynomial of degree at most 2n − 1.Since we are interested in whether or not y j (0, x; c) ∈ L 2 ((0, d); dx) for some d ∈ (0, ∞), the presence of logarithmic terms is irrelevant and the deciding L 2 -criterion for solutions of τ 2n (c)y(µ, • ; c) = 0 simply becomes In conclusion, to settle the essential self-adjointness problem (1.6) one needs to establish the existence of Turning briefly to the content of each section, we note that Section 2 introduces minimal and maximal operators associated with general differential expressions τ 2n of order 2n, n ∈ N, in L 2 ((0, ∞); 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 τ 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 τ 2 (c) and τ 4 (c) are treated explicitly.Properties of the (real part of the) roots α j (c) of D 2n ( • ; c) = 0 are the center piece of our principal Section 4, culminating in Theorem 4.5 and Corollary 4.6 which settle the essential self-adjointness problem (1.6).The techniques involved are related to the Grace-Haewood theorem [36, p. 126], the Routh-Hurwitz criterion, and Orlando's formula [12,§ XV.7].Appendix A shows with the help of Galois theory that c 6 , c 7 are algebraic numbers that cannot be expressed as radicals over Q; we conjecture this actually remains the case for all c n , n ∈ N, n ≥ 6.
Finally, some remarks on the notation employed: We denote by C M ×N , M, N ∈ N, the linear space of M × N matrices with complex-valued entries.I N represents the identity matrix in C N .The spectrum of a matrix (or closed operator in a Hilbert space) T is denoted by σ(T ).The abbreviation N 0 = N ∪ {0} is used.
and then also obtains ..
T 2n,min = T 2n,min . (2.10) one infers for (d, e) ⊂ (a, b) Lagrange's identity via integrations by parts (2.12) Moreover, if u(µ, • ) and v(µ, • ) are solutions of Finally, we also recall the known fact, dom(T 2n,min ) = {g ∈ dom(T 2n,max ) | for all h ∈ dom(T 2n,max ): In the following, the number of L 2 ((a, b); rdx)-solutions u(µ ± , • ) of is denoted by n ± (T 2n,min ) and called the deficiency indices of T 2n,min .This notion is well-defined as n ± (T 2n,min ) is known to be constant throughout the open complex upper and lower half-plane.As a result, one typically chooses µ ± = ±i.Since the coefficients of τ 2n are real-valued, one obtains by a result of von Neumann [44] that  All other cases, where 1 ≤ n ± (T 2n,min ) ≤ 2n, describe various degrees of limit circle cases of τ 2n , with n ± (T 2n,min ) = 2n representing the extreme case.⋄ In the bulk of this paper we are particularly interested in the special case where denoting the associated (pre)minimal and maximal operators in L 2 ((0, ∞); dx) by T 2n,min (c), .
T 2n,min (c), T 2n,max (c), etc.In particular, we are interested in the question, "for which values of c ∈ R is T 2n,min (c) self-adjoint resp., ..

Perturbed Euler Differential Systems and Their Deficienciy Indices
In this section we will prove that it suffices to focus on the spectral parameter µ = 0 when trying to determine the number of L 2 ((0, d); dx)-solutions y(µ, • ) of for fixed d ∈ (0, ∞) (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, • ), that is, one can reduce (3.1) to the far simpler case µ = 0.
In the following ) and we will study first-order systems of differential equations of the particular form where Y ( • ) represents either an N × 1 solution vector or an N × N solution matrix, N ∈ N, which generally is multi-valued, and 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 with fundamental (typically, many-valued) matrix solutions of the form where ) 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 × r, 1 ≤ r ≤ N , sizes, denoted by J r (α q ).In particular, if J r (α q ) is of the form then explicitly demonstrating the appearance of powers of logarithms of ζ in (3.5) in the case where A 0 has an eigenvalue α q whose algebraic multiplicity strictly exceeds its geometric one.In particular, the eigenvalues α q of A 0 are determined via the characteristic equation for A 0 , also called the indicial equation, The general, or perturbed, Euler case (3.2) leads to analogous results as follows.
In this context we also refer to Sections 4.3, 4.4, particularly, Theorem 4.11, in Teschl [43], 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 σ(A 0 ) ∩ {σ(A 0 ) + Z} = ∅, (3.11) it was proven by Fuchs [11] (cf. Hille [20, Theorem 9.5.1]) that the perturbed Euler differential system (3.2) has fundamental matrix solutions of the form where again C ∈ C N ×N is nonsingular.
The case where the spectral assumption (3.11) on A 0 is violated is much more involved 1 .What follows is a shortened description of Hille [20, 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 [20, p. 342-352].
We conclude this overview by specializing the 1st-order N × N perturbed Euler system (3.2) to the N th-order scalar case (a special case of which is depicted in (3.1)).Consider the scalar N th-order differential equation 1 In fact, we quote Hille [20, p. 344] in this context: ". . .A number of arguments are available in the literature all of them more or less corny.What I shall give here is not the corniest; . . ." where the coefficients b j ( • ), 0 ≤ j ≤ N − 1, are of the form with a j ( • ) entire (resp., analytic in D(0; R)).The scalar ODE (3.14) subordinates to the perturbed Euler differential system (3.2) upon identifying A(ζ) with the N × N matrix , where the solutions with y j ( • ), 1 ≤ j ≤ N , linearly independent solutions of (3.14).In this scalar context the matrix 3) is thus of the form and hence the eigenvalues α q of A 0 prominently figuring in the solution (3.10) are determined via the indicial equation (3.9), D N ( • ) = 0, where Given these results we can return to the half-line differential expression τ 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) In this case the indicial equation further reduces to Thus, we can state the following result.
Moreover, the deficiency indices n ± (T 2n,min (c)) (with T 2n,min (c) representing the closure of and hence In particular, (3.26) Proof.The µ-independence of # L 2 τ 2n (c)| (0,d) follows from the structure of the solutions Y j in (3.10), the fact that for each d ∈ (0, ∞), the power x α lies in L 2 ((0, d); dx) if and only if Re(α) > −1/2, independently of the presence of any logarithmic factors, and finally that only the spectrum of A 0 determines the powers α q in (3.10).Since c ∈ R, τ 2n (c) possesses an anti-unitary conjugation operator (effected by complex conjugation of elements in L 2 ((0, ∞); dx)) and one obtains by (2.17), Moreover by a special case of Kodaira's decomposition principle (2.20) for deficiency indices, where P ( • ) is any polynomial, then for all d ∈ (0, ∞), In particular, at z = 0 one obtains Consequently, for c = −[(2n − 1)!!] 2 2 2n one has a double zero at α = k − (1/2) and there are two solutions of the type in this case.⋄ Next, we now recall the special situation n = 1 which is explicitly solvable for general spectral parameter µ in terms of Bessel functions as follows: Example 3.5.Assuming the case n = 1 in (3.1) we consider (3.36) The associated characteristic equation has the following two complex-valued solutions choosing the principal branch for [ • ] 1/2 with branch cut (−∞, 0], such that With this convention in place one checks that for all c ∈ R, one has the ordering, Then the nonhomogenous differential equation (3.36) has the following fundamental system of solutions (cf.[1, No. 9.1.49,p. 362]) (Thus, γ(c) ∈ {[0, ∞)\N 0 } ∪ i(0, ∞) in the generic case.) ) In the exceptional case, where γ(c) = k ∈ N 0 , one obtains Here J κ ( • ) represent the standard Bessel functions of order κ ∈ C and first kind, and Y k ( • ) denotes the Bessel function of order k ∈ N 0 and second kind (see, e.g., [1,Ch. 9]).Moreover, one verifies (cf.[1, p. 360]) that (here W (f, g) = f g ′ − f ′ g denotes the Wronkian of f and g), and that the fundamental system of solutions y 1 (µ, •; c), y 2 (µ, •, c) (3.42), (3.48) of (3.36) is entire with respect to µ ∈ C for fixed x ∈ (0, ∞), and real-valued for µ ∈ R.
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 Re(α j (c) > −1/2, j = 1, 2, one concludes that Here 0 F 1 b 1 • represents the generalized hypergeometric function given by with (a) k denoting Pochhammer's symbol, In the exceptional case, where γ(c) = k ∈ N 0 , one obtains • , is given by a Mellin-Barnes-type integral, where C is a contour beginning and ending at +∞ encircling all poles of Γ(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 [13].) ⋄ For details on generalized hypergeometric functions and Meijer's G-function we refer, for instance, to [4], [8, Ch.IV, Sects. 5. (3.58) The associated characteristic equation has the following four complex-valued solutions, employing the principal branch (3.39) for [ • ] 1/2 .With this convention, one checks that for all c ∈ R, one has Then the nonhomogenous differential equation (3.58) has the following fundamental system of solutions, (3.63) Asymptotically, and thus, the four functions are indeed linearly independent. Here • represents the generalized hypergeometric function given by That these functions are in fact solutions of (3.58) can be confirmed by direct verification using the differential equation for generalized hypergeometric functions.
and a fundamental system of solutions is given by, (3.73) Asymptotically, • , is again given by a Mellin-Barnestype integral, where C is a contour beginning and ending at +∞ encircling all poles of Γ(c j − •), 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.
and a fundamental system of solutions is given by, Asymptotically, and a fundamental system of solutions is given by, (3.80) Asymptotically, One observes that the case c = −9/16, is again precisely the borderline of semiboundedness of T min,4 (c) again in accordance with (1.9) (see also (1.10)) and Remark 3.4. Asymptotically, Once more, as µ → 0, the fundamental system of solutions of (3.58) considerably simplifies to x ∈ (0, ∞).

On the Real Part of the Roots of
For n ∈ N and c ∈ R, let D 2n ( • ; 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 ( • ; 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 ∈ R, let the roots of D 2n ( • ; c) = 0 be denoted α j (c), j = 1, . . ., 2n.By the continuous dependence of the roots of a polynomial on the coefficients (see [31,Theorem (1.4)]), we may choose our labelling such that each α j (c) is a continuous function of c and Re(α Note that Re(α j (0)) = α j (0) = j − 1 for j = 1, . . ., 2n.The fact that will be of crucial importance in all that follows.
Example 4.1.Figure 1 shows the graphs of the the real parts of the roots of D 6 ( • ; c) as functions of c ∈ R. The scale for the x-axis has been chosen such that x = c 1/6 for c > 0 and x = sgn(c)|c| 1/6 for c < 0. The dotted lines show the graphs of the real parts of the roots of ( • ) 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 → ±∞, the graph of each function Re(α j (c)) approaches one of these straight lines asymptotically.One observes that for c ≪ 0, one has Re(α As will be shown later, we have where the algebraic numbers on the right are roots of the quadratic equation 27c 2 − 958720c − 677376000 = 0.If j ∈ {4, 5, 6}, then Re(α j (c)) > −1/2 for all c ∈ R.
Before we prove part (a), we recall that by the continuity argument given in the first paragraph of this proof, for every 1 ≤ j ≤ n − 1, there exists some c < 0 such that Re(α j (c)) = −1/2.By our observations above, this c must be one of the constants c (k) n with 1 ≤ k ≤ ⌊n/2⌋.Similarly, for every 1 ≤ j ≤ n, there exists some c > 0 such that Re(α j (c)) = −1/2 and, by our observations above, this c must be one of the constants c In the case when n is even, the argument is, mutatis mutandis, the same.The situation is summarized in Figure 2

.51) Remark 3 . 6 .
In view of the next example, where n = 2, in fact, in view of the general case n ∈ 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 ∈ R is such that [α 1 (c) − α 2 (c)]/2 ̸ ∈ Z, the nonhomogenous differential equation (3.36) has the following fundamental system of solutions

When applied to the two roots z 1 Lemma 4 . 3 .
, z 2 of D 2n ( • ; 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 ∈ C | Im(z) = y 0 }, where y 0 := [Im(z 1 ) + Im(z 2 )]/2 > 0. Now recall that by (4.6) all the critical points of D 2n ( • ; c) are real.Thus, the closed half-plane {z ∈ C | Im(z) ≥ y 0 } would be devoid of any critical points of D 2n ( • ; c).This is the desired contradiction.□The second lemma is concerned with the asymptotic behavior of the real parts of the roots of D 2n ( • ; c) as c → ±∞.For j ∈ {1, 2, . . ., 2n} and c ∈ R,