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

Abstract

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.

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 $\mathbb {R}$:

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

(8.1)

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

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

(8.2)

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

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

(8.3)

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,

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

(8.4)

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,

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

(8.5)

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

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

(8.6)

The proof falls out of the identity

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

(8.7)

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

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

(8.8)

which yields

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

(8.9)

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.