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Bound states in soft quantum layers

David Krejcirik, Jan Kriz

Abstract

We develop a general approach to study three-dimensional Schroedinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut locus in the Euclidean space. If the surface is asymptotically planar in a suitable sense, we give an estimate on the location of the essential spectrum of the Schroedinger operator. Moreover, if the surface coincides up to a compact subset with a surface of revolution with strictly positive total Gauss curvature, it is shown that the Schroedinger operator possesses an infinite number of discrete eigenvalues.

Bound states in soft quantum layers

Abstract

We develop a general approach to study three-dimensional Schroedinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut locus in the Euclidean space. If the surface is asymptotically planar in a suitable sense, we give an estimate on the location of the essential spectrum of the Schroedinger operator. Moreover, if the surface coincides up to a compact subset with a surface of revolution with strictly positive total Gauss curvature, it is shown that the Schroedinger operator possesses an infinite number of discrete eigenvalues.
Paper Structure (4 sections, 8 theorems, 85 equations, 3 figures)

This paper contains 4 sections, 8 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Parallel coordinates
  3. Rotationally symmetric layers
  4. The proofs

Key Result

Theorem 1

Let $\Sigma$ be an orientable smooth surface which is asymptotically cut-locus planar Ass.planar.cut and admits an integrable Gauss curvature Ass.K. Let the tubular neighbourhood tube do not overlap itself with some positive $a$, i.e.Ass.local and Ass.global hold. Let $V$ be an essentially bounded f Moreover, if $\Sigma$ coincides up to a compact subset with a cylindrically symmetric surface with

Figures (3)

  • Figure 1: The piecewise smooth curve \ref{['curve.special']} (symmetrically extended to $s \in \mathbb{R}$) and its cut-locus (red).
  • Figure 2: Towards the proof of Conjecture \ref{['Conj']}. Surface (a) is the curve of Figure \ref{['Fig.KKK']} with $\theta=\pi$ and $R=1$ translated along axis $x^3$. Then it is not asymptotically cut-locus planar because the distance between the flat parts equals the constant $2$ as $x^2 \to \infty$. Surface (b) is obtained from (a) by taking the radius $R$ in \ref{['curve.special']} dependent both on $s$ and $x^3$, namely $R(s,x^3) := 1+(x^3)^2/(1+s^2)$. Then (b) is not asymptotically cut-locus planar either (the distance between the modified flat parts remains $2$ as $x^2 \to \infty$ and $x^3=0$), while $K,M \to 0$ as $x^2 \to \infty$ and $x^3$ is fixed (the challenge is to have \ref{['Ass.planar']} globally). Surface (c) is the curve of Figure \ref{['Fig.KKK']} with $\theta = \frac{5}{7} \pi$ and $R$ as for (b); then $c_\pm \to \infty$ as $x^2 \to \infty$ and $x^3$ is fixed.
  • Figure 3: The geometry of the generating curve $\Gamma$.

Theorems & Definitions (17)

  • Theorem 1
  • Conjecture 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 7 more