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Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials

Martin Averseng, Nicolas Frantz, Frédéric Hérau, Nicolas Raymond

TL;DR

This work extends semiclassical tunneling analysis to a non-selfadjoint Schrödinger operator with a complex-valued potential, L(h) = -h^2 d_x^2 + e^{iα}V with V even and possessing two symmetric wells. The authors develop a robust framework combining elliptic estimates, Agmon localization, and WKB constructions to reduce spectral analysis to the left and right wells and to a complex harmonic oscillator model. They demonstrate that the low-energy spectrum near the origin consists of exponentially close pairs, with gaps governed by an explicit complex tunneling factor S(α) and a prefactor A, while all other spectrum remains separated by a larger O(h) scale; for α ≠ 0 the pair exhibits a characteristic rotation in the complex plane. The analysis yields a precise asymptotic description of the eigenvalue gap and the associated quasi-modes, providing a rigorous extension of the standard HS84-type tunneling results to non-selfadjoint settings with complex potentials. This has implications for understanding tunneling and decay for non-selfadjoint quantum systems and related PDEs in the semiclassical regime.

Abstract

We consider the non-selfadjoint, semiclassical Schrödinger operator $\mathscr{L}(h) := -h^2\partial_x^2+e^{iα}V$, where $α\in (-π,π)$ and $V: \mathbb{R}\to \mathbb{R}_+$ is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of $\mathscr{L}(h)$ near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a $\mathscr{O}(e^{-S/h})$ distance where $S > 0$ is explicit), each pair being separated from the others by a distance $\mathscr{O}(h)$. A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when $α\neq 0$, they quickly rotate around each other as $h$ goes to $0$.

Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials

TL;DR

This work extends semiclassical tunneling analysis to a non-selfadjoint Schrödinger operator with a complex-valued potential, L(h) = -h^2 d_x^2 + e^{iα}V with V even and possessing two symmetric wells. The authors develop a robust framework combining elliptic estimates, Agmon localization, and WKB constructions to reduce spectral analysis to the left and right wells and to a complex harmonic oscillator model. They demonstrate that the low-energy spectrum near the origin consists of exponentially close pairs, with gaps governed by an explicit complex tunneling factor S(α) and a prefactor A, while all other spectrum remains separated by a larger O(h) scale; for α ≠ 0 the pair exhibits a characteristic rotation in the complex plane. The analysis yields a precise asymptotic description of the eigenvalue gap and the associated quasi-modes, providing a rigorous extension of the standard HS84-type tunneling results to non-selfadjoint settings with complex potentials. This has implications for understanding tunneling and decay for non-selfadjoint quantum systems and related PDEs in the semiclassical regime.

Abstract

We consider the non-selfadjoint, semiclassical Schrödinger operator , where and is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a distance where is explicit), each pair being separated from the others by a distance . A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when , they quickly rotate around each other as goes to .

Paper Structure

This paper contains 30 sections, 42 theorems, 246 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Context and motivation
  4. Outline of the proof of Theorem \ref{['thm:gap']}
  5. Notation
  6. An elliptic estimate and its application to exponential decay
  7. The elliptic estimate
  8. Exponential localization of eigenfunctions of the simple-well operator
  9. Simple-well operator
  10. Agmon distance and exponential localization
  11. Quasimodes of the double-well operator and its quadratic approximation
  12. Resolvent of the simple-well operator
  13. The complex harmonic oscillator
  14. Nature and a priori location of the spectrum
  15. Resolvent bounds for $\mathscr{L}_\ell(h)$
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $V : \mathbb{R} \to \mathbb{R}_+$ be smooth, even and bounded-below at infinity, i.e. Suppose that $V$ vanishes exactly at two points $x_\ell$ and $x_r= -x_\ell$, and that these minima are non-degenerate, i.e., $V"(x_\ell) = V"(x_r) > 0$. Given $\alpha \in (\pi,\pi)$, let $\mathscr{L}(h)$ be the unbounded operator on $L^2(\mathbb{R})$ with domain $H^2(\mathbb{R})$, defined by There exists $h

Figures (2)

  • Figure 1: Graph of a function $V$ satisfying the assumptions (black, solid line) and the potential $V_\ell = V + \Sigma_\ell$ (red, dashed line) for the simple well problem and $V_\ell^{[2]}$ the Taylor approximation of $V_\ell$ to order $2$ at $x_\ell$.
  • Figure 2: The cutoff functions $\chi_\ell^+$ and $\underline{\chi_\ell}^+$ in Corollary \ref{['cor:quasimodes']}.

Theorems & Definitions (81)

  • Theorem 1.1
  • Remark 1.2: Comments on Theorem \ref{['thm:gap']}
  • Definition 2.1: $\kappa$-subsolution
  • Proposition 2.2: Elliptic estimate
  • proof
  • Lemma 2.3: Decay of the Agmon distance
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6: The Agmon distance is a subsolution
  • proof
  • ...and 71 more