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Inequalities à la Pólya for the Aharonov--Bohm eigenvalues of the disk

Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher

Abstract

We prove an analogue of Pólya's conjecture for the eigenvalues of the magnetic Schrödinger operator with Aharonov--Bohm potential on the disk, for Dirichlet and magnetic Neumann boundary conditions. This answers a question posed by R. L. Frank and A. M. Hansson in 2008.

Inequalities à la Pólya for the Aharonov--Bohm eigenvalues of the disk

Abstract

We prove an analogue of Pólya's conjecture for the eigenvalues of the magnetic Schrödinger operator with Aharonov--Bohm potential on the disk, for Dirichlet and magnetic Neumann boundary conditions. This answers a question posed by R. L. Frank and A. M. Hansson in 2008.
Paper Structure (3 sections, 11 theorems, 76 equations, 4 figures, 1 table)

This paper contains 3 sections, 11 theorems, 76 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm:polyad']}
  3. Proof of Theorem \ref{['thm:polyan']}

Key Result

Theorem 1.1

Let $\alpha\in\left[0,\frac{1}{2}\right]$. Then

Figures (4)

  • Figure 1: Some zeros of Bessel functions as functions of $\alpha$.
  • Figure 2: Some zeros of derivatives of Bessel functions as functions of $\alpha$.
  • Figure 3: The functions $q_1(c)$ and $q_2(c)$.
  • Figure 4: The difference $\mathcal{Q}(\lambda)-\frac{\lambda^2}{4}$ plotted as a function of $\lambda\in\left[\frac{5}{2}, 9\right]$. The plot has been produced using floating-point arithmetic and is included for illustration only: it does not constitute a part of the proof.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: FLPS
  • Remark 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lem:g']}
  • ...and 13 more