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Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition


We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to \(-\,\infty \), a narrow cluster of finitely many eigenvalues tends to \(-\,\infty \), while the eigenvalues above this cluster remain bounded from below. Certain “rogue” eigenvalues break away from this cluster and tend even faster toward \(-\,\infty \). The spectrum can be visualized as the intersection points of two objects in the plane—a spiral curve depending on the Schrödinger potential, and a set of curves depending on the branching factor, the diameter of the tree, and the Robin parameter.

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Appendix on Moments

Appendix on Moments

By Favard’s Theorem [14, Theorem 4.4], \(\{P_n\}\) is a sequence of orthogonal polynomials with respect to a measure \(\mathrm{d}\psi \), with \(\psi \) being a (not strictly) increasing function on \(\mathbb {R}\):

$$\begin{aligned} \int P_n(v)P_m(v)\,\mathrm{d}\psi (v) = 0 \qquad (m\not =n). \end{aligned}$$

For each \(n>0\), let the positive numbers \(A_{n1},\ldots , A_{nn}\) be the Gaussian quadrature weights, and define the functions

$$\begin{aligned} \psi _n(v)= \left\{ \begin{array}{l@{\quad }l@{\quad }l} 0 &{}\;\;\text { if } v< v_{n1}&{} \\ A_{n1}+A_{n2}+ \cdots +A_{np}&{}\;\;\text { if }v_{np} \le \;\; v<v_{n, p+1}&{}(1\le p<n).\\ \mu _0 &{}\;\;\text { if } v\ge v_{nn}&{} \end{array} \right. \end{aligned}$$

The moments of \(\psi \) and its approximants \(\psi _k\) are

$$\begin{aligned} \mu _k = \int v^k\,\mathrm{d}\psi (v), \qquad \mu _k^{(n)} = \int v^k\,\mathrm{d}\psi _n(v). \end{aligned}$$

Since the roots of \(P_n\) lie between \(-(b+1)\) and \((b+1)\),

$$\begin{aligned} \mathrm {supp}(\mathrm{d}\psi _n) \subset [-(b+1),b+1], \qquad \mathrm {supp}(\mathrm{d}\psi ) \subset [-(b+1),b+1]. \end{aligned}$$

The \(\mathrm{d}\psi _n\) approximate \(\mathrm{d} \psi \) as measures in the sense that \(\mathrm{d}\psi _n\) produces integrals of polynomials of degree \(k \le 2n-1\) exactly. Therefore,

$$\begin{aligned} \mu ^{(n)}_k=\mu _k, \qquad 0\le k \le 2n-1. \end{aligned}$$

Since \(P_n(v)\) and \(Q_n(v)\) are either even or odd, their roots are symmetric about the origin, and thus all odd moments vanish,

$$\begin{aligned} \mu _k=0 \;\;\text { and }\;\; \mu ^{(n)}_k=0 \qquad (k\text { odd}). \end{aligned}$$

The series for the ratio (5.16) used in the proof of Theorem 9 can be refined as follows.

Proposition 13

The ratio of \(P_{n+1}(v)\) and \(Q_{n+1}(v)\) satisfies

$$\begin{aligned} \dfrac{-bP_{n+1}(v)}{vQ_{n+1}(v)} = 1-v^{-2}\sum \limits _{j=0}^{\infty }v^{-j}\mu _j^{(n)}. \end{aligned}$$

The proof falls out of the identity

$$\begin{aligned} \frac{-bP_{n+1}(v)}{vQ_{n+1}(v)} = 1-\dfrac{1}{v} \int \dfrac{d \psi _n(t)}{v-t}, \end{aligned}$$

which comes from \(-bP_n(v)=Q_{n+1}(v)\) (Proposition 5) and [14, Chapter III, Theorem 4.3].

This leads to a refinement of the asymptotics of the rogue curves in Theorem 9. See [30] for details.

Theorem 14

If \(b\ge 2\) and \(\alpha \not =0\), the components \({{\mathcal {C}}}_n^n\) and \({{\mathcal {C}}}_n^{n-1}\) of \(D_n(y,z)=0\) and the component \(\mathring{{\mathcal {C}}}_n^n\) of \(\mathring{D}_n(y,z)=0\) have the following asymptotic behavior as \(y\rightarrow \infty \) in the yz-plane.

$$\begin{aligned} \alpha z= -cy+cy^{-1}+c^{(n)}_{-2}y^{-2}+c^{(n)}_{-3}y^{-3}+c^{(n)}_{-4}y^{-4}+ \cdots + c^{(n)}_{-k}y^{-k}+ \cdots \end{aligned}$$

The coefficient \(c^{(n)}_{-k}\) depends on \(\alpha \), b, and \(\{\mu _j^{(n-1)}\}_{j=0}^{k-1}\) only.

Putting this together with (8.4) says that the coefficient \(c^{(n)}_{-k}\) stabilizes when n is large enough that \(k < 2n\). Furthermore, the expansions for the curves \({{\mathcal {C}}}^n_n\), etc. for two different values of n are different.

The measure \(\mathrm{d}\mu \) can be computed easily from the expressions (5.4) of \(P_n\) and \(Q_n\). As a density, it is the limit of the density of roots of these polynomials as \(n\rightarrow \infty \). By putting \(\xi =\exp (i\theta )\) we obtain \(-b^{-(n+1)/2}Q_n=\sin n\theta /\sin \theta \). Thus we seek the density of roots of \(\sin n\theta \) as a function of v, which is

$$\begin{aligned} R_n(v) = \sin \Big ( n\arccos \frac{v}{2b^{1/2}} \Big ), \qquad \left| v \right| < 2b^{1/2}, \end{aligned}$$

which yields

$$\begin{aligned} \mathrm{d}\mu (v) = \frac{2b^{1/2}\,\mathrm{d}v}{\pi \sqrt{4b-v^2}}, \qquad \left| v \right| < 2b^{1/2}. \end{aligned}$$

The following proposition refines the expression of \(P_n\) and \(Q_n\); its proof is omitted (see [30]).

Proposition 15

For \(n\ge 1\), \(P_n(v)\) is the polynomial part of

$$\begin{aligned}&v^n - (n-1)b\, v^{n-2} + \dfrac{(n-3)(n-2)}{2}b^2v^{n-4}-\dfrac{(n-5)(n-4)(n-3)}{6}b^3v^{n-6}\\&\quad + \cdots , \end{aligned}$$

and \(Q_n(v)\) is the polynomial part of

$$\begin{aligned}&-b\,v^{n-1} + (n-2)b^2\, v^{n-3} \\&\quad - \dfrac{(n-4)(n-3)}{2}b^3v^{n-5}+\dfrac{(n-6)(n-5)(n-4)}{6}b^4v^{n-7}+ \cdots , \end{aligned}$$

in which the ellipses indicate lower-degree monomials.

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Hess, Z.W., Shipman, S.P. Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition. Ann. Henri Poincaré 22, 2531–2561 (2021).

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Mathematics Subject Classification

  • 34B45
  • 34B08
  • 34A26
  • 34L15
  • 34L20
  • 34L05