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2D Schrödinger operators with singular potentials concentrated near curves

Yuriy Golovaty

TL;DR

The paper analyzes 2D Schrödinger operators $H_\varepsilon=-\Delta+W+V_\varepsilon$ with potentials $V_\varepsilon$ supported near a smooth curve $\gamma$, showing that as $\varepsilon\to0$ the spectrum and eigenfunctions converge to a limit problem whose form depends on whether the 1D model $-d^2/dr^2+V$ has a zero-energy resonance. In the resonant case, the limit is a curved-interface problem for $\mathcal{H}=-\Delta+W$ with explicit transmission conditions on $\gamma$ involving the curvature $\varkappa$, a slope parameter $\theta=h(+\infty)/h(-\infty)$, and a curvature-weighted integral $\mu$, while in the non-resonant case the limit decouples into a direct sum $\mathcal{D}^-\oplus\mathcal{D}^+$ with Dirichlet-type transmission on $\gamma$. The authors construct inner-outer asymptotics, derive solvability conditions, and build quasimodes to prove that eigenvalues of $H_\varepsilon$ converge to eigenvalues of the appropriate limit operator with rate $O(\varepsilon)$; they also discuss potential accumulation to $-\infty$ in certain regularizations. The results reveal a deep connection between spectral behavior and the geometry of the interaction curve, providing solvable models for singular interactions supported on curves in the plane.

Abstract

We investigate the Schrödinger operators $H_\varepsilon=-Δ+W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials $V_\varepsilon$ which are localized around a smooth closed curve $γ$. The operators $H_\varepsilon$ can be viewed as an approximation of the heuristic Hamiltonian $H=-Δ+W+a\partial_νδ_γ+bδ_γ$, where $δ_γ$ is Dirac's $δ$-function supported on $γ$ and $\partial_νδ_γ$ is its normal derivative on $γ$. Assuming that the operator $-Δ+W$ has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The transmission conditions on $γ$ for the eigenfunctions $u^+=αu^-$, $α\, \partial_νu^+-\partial_νu^-=βu^-$, which arise in the limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral properties of $H_\varepsilon$ and the geometry of $γ$.

2D Schrödinger operators with singular potentials concentrated near curves

TL;DR

The paper analyzes 2D Schrödinger operators with potentials supported near a smooth curve , showing that as the spectrum and eigenfunctions converge to a limit problem whose form depends on whether the 1D model has a zero-energy resonance. In the resonant case, the limit is a curved-interface problem for with explicit transmission conditions on involving the curvature , a slope parameter , and a curvature-weighted integral , while in the non-resonant case the limit decouples into a direct sum with Dirichlet-type transmission on . The authors construct inner-outer asymptotics, derive solvability conditions, and build quasimodes to prove that eigenvalues of converge to eigenvalues of the appropriate limit operator with rate ; they also discuss potential accumulation to in certain regularizations. The results reveal a deep connection between spectral behavior and the geometry of the interaction curve, providing solvable models for singular interactions supported on curves in the plane.

Abstract

We investigate the Schrödinger operators in with the short-range potentials which are localized around a smooth closed curve . The operators can be viewed as an approximation of the heuristic Hamiltonian , where is Dirac's -function supported on and is its normal derivative on . Assuming that the operator has only discrete spectrum, we analyze the asymptotic behaviour of eigenvalues and eigenfunctions of . The transmission conditions on for the eigenfunctions , , which arise in the limit as , reveal a nontrivial connection between spectral properties of and the geometry of .

Paper Structure

This paper contains 10 sections, 10 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of Problem and Main Results
  3. Preliminaries
  4. Formal Asymptotics
  5. Case of zero-energy resonance
  6. Non-resonant case
  7. Quasimodes of $H_\varepsilon$
  8. Proof of Main Results
  9. Proof of Theorem \ref{['MainThrmRes']}
  10. Proof of Theorems \ref{['MainThrmNoRes']}

Key Result

Theorem 1

Assume that the operator $-\frac{d^2}{d r^2}+V$ in $L^2(\mathbb R)$ possesses a zero-energy resonance with the half-bound state $h$. (i) Suppose that $\{\lambda^\varepsilon\}_{\varepsilon\in\mathcal{E}}$ is a sequence of eigenvalues of $H_\varepsilon$ and $\{u_\varepsilon\}_{\varepsilon\in\mathcal{E as $\mathcal{E}\ni\varepsilon\to 0$, and $u$ is a non-zero function, then $\lambda$ is an eigenvalu

Figures (3)

  • Figure 1: The $\partial_\nu\delta_\gamma$-like potential (part of the plot is cut out for better visualization).
  • Figure 2: The local coordinates in $\omega_\varepsilon$.
  • Figure 3: Plot of the function $\zeta$.

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3: -0.1ptPDEVinitiSpringer
  • Lemma 1
  • ...and 9 more