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Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

Gregory Berkolaiko, Graham Cox, Jeremy L. Marzuola

Abstract

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schrödinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

Abstract

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schrödinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.

Paper Structure

This paper contains 8 sections, 3 theorems, 35 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The spectral flow
  3. The one-dimensional case
  4. The rectangle
  5. Example for a Rectangle
  6. $\{ Z_k \} = \{ \frac{1}{2} \ell \}$
  7. $\{ Z_k \} = \{ \frac{1}{3} \ell, \frac{2}{3} \ell \}$
  8. An example with nodal deficiency $3$ on the rectangle

Key Result

Lemma 1

For $\epsilon$ sufficiently small, the value $-\sigma$ is an eigenvalue of $\Lambda_+ (\epsilon) + \Lambda_- (\epsilon)$ if and only if $\lambda_* + \epsilon = \gamma_k(\sigma)$ for some $k \in \mathbb{N}$.

Figures (6)

  • Figure 1: Numero-analytic solution of the spectral flow on the tetrahedron quantum graph (left) and on a rectangle (right), as described in Appendix \ref{['app:rec']}. In both cases the number of curves crossing $\lambda_*+\epsilon$ matches the nodal deficiency (2 on the left and 3 on the right).
  • Figure 2: Illustrating the result of Observations \ref{['obs:rectangle1']} and \ref{['obs:rectangle2']}. For the simple eigenvalue (A), the nodal deficiency is 4, which equals the Morse index of $\Lambda_+ + \Lambda_-$. The degenerate eigenvalue (B) also has nodal deficiency 4. The point $(6,2)$ generates an additional negative eigenvalue of $\Lambda_+ + \Lambda_-$ but does not contribute to the nodal deficiency.
  • Figure 3: Suspected intersection of flow curves for a quantum graph and its zoom.
  • Figure 4: The behavior of the first four eigenvalues of $L_\sigma$ in one dimension, with $k_* = 4$. The fourth eigenvalue, $\lambda_4(\sigma) = \lambda_*$, is constant, whereas the first three strictly increase to $\lambda_*$, and the fifth converges to some number strictly greater than $\lambda_*$, as claimed in \ref{['1Dlimita']} and \ref{['1Dlimitb']}.
  • Figure 5: The behavior of $\gamma_k(\sigma)$ as $\sigma\to\infty$. The dashed curve has $\gamma_{k}(0) = \lambda_{mn} < \lambda_*$ and $\gamma_k(\infty) > \lambda_*$, and hence generates a negative eigenvalue $-\sigma_0$ for $\Lambda_+ (\epsilon) + \Lambda_- (\epsilon)$. The other two eigenvalues curves correspond to $(m,n) \leq (m_*, n_*)$, and hence stay below $\lambda_*$ for all finite values of $\sigma$.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Definition 2
  • Theorem 1
  • proof