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Weak coupling for Schrödinger operators with complex potentials

Jussi Behrndt, Markus Holzmann, Petr Siegl, Nicolas Weber

TL;DR

This work extends the classical weak coupling phenomenon for Schrödinger operators to non-self-adjoint settings with complex potentials in 1D and 2D. By leveraging the Birman-Schwinger principle and a holomorphic reduction, the authors derive necessary and sufficient conditions for the existence, uniqueness, and multiplicity of discrete eigenvalues emerging from the essential spectrum, including explicit asymptotic formulas in both dimensions. The analysis hinges on a careful decomposition of the Birman-Schwinger operator, a transformation that reduces the problem to zeros of a holomorphic function, and a Rouche-based argument to isolate a single simple zero in the weak-coupling limit. The results reveal a qualitative difference between 1D and 2D, show sharp spectral enclosures, and connect to Rollnik-type potentials, offering a robust non-self-adjoint generalization of Simon’s classical weak coupling theory.

Abstract

We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].

Weak coupling for Schrödinger operators with complex potentials

TL;DR

This work extends the classical weak coupling phenomenon for Schrödinger operators to non-self-adjoint settings with complex potentials in 1D and 2D. By leveraging the Birman-Schwinger principle and a holomorphic reduction, the authors derive necessary and sufficient conditions for the existence, uniqueness, and multiplicity of discrete eigenvalues emerging from the essential spectrum, including explicit asymptotic formulas in both dimensions. The analysis hinges on a careful decomposition of the Birman-Schwinger operator, a transformation that reduces the problem to zeros of a holomorphic function, and a Rouche-based argument to isolate a single simple zero in the weak-coupling limit. The results reveal a qualitative difference between 1D and 2D, show sharp spectral enclosures, and connect to Rollnik-type potentials, offering a robust non-self-adjoint generalization of Simon’s classical weak coupling theory.

Abstract

We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schrödinger operators with complex-valued -potentials in a weak coupling regime. We derive necessary and sufficient conditions on the potential for the existence or absence of discrete eigenvalues in this regime and also analyze their uniqueness and algebraic multiplicity. Our results can be viewed as natural non-self-adjoint extensions of the well-known classical weak coupling phenomenon for self-adjoint Schrödinger operators with real-valued potentials going back half a century to Simon's famous paper [Simon 1976].

Paper Structure

This paper contains 29 sections, 15 theorems, 238 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Strategy of the proof
  3. Notations and conventions
  4. The Birman-Schwinger principle and eigenvalues in the weak coupling regime
  5. Birman-Schwinger principle
  6. Reduction to zeros of a holomorphic function
  7. Transformation of Lambda
  8. Characterization of all eigenvalues for Rollnik-type potentials
  9. Proofs of the main results
  10. Proof of Theorem \ref{['theorem:s=2']}
  11. Proof of (i)
  12. Proof of (ii)
  13. Proof of Corollary \ref{['cor:s=2.V.real']} for d=1
  14. Proof of (i) for d=1
  15. Proof of (ii) for d=1
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $H_\beta = -\Delta -\beta V$ be as in H.beta.def with $V$ satisfying assumption:family_v and def:Rollnik_space_mainjussi. For $U$ and $U_1$ in def:U_U_1_intro assume $U \neq 0$ and define where $\gamma$ is the Euler-Masceroni constant. Then there exist $\varepsilon > 0$ and $R > 0$ such that for every $\beta\in \mathbb{C}$ with $|\beta|<\varepsilon$ the following assertions hold.

Figures (2)

  • Figure 1: A schematic plot of Corollary \ref{['cor:s=2.V.real']} for $U = U_1 = R' = 1$. In (a) and (c) the coupling parameter $\beta\rightarrow 0$ (along three different curves) leads to a simple discrete eigenvalue $\lambda_\beta$ of $H_\beta$ in (b) and (d).
  • Figure 2: An illustration of the set $\Omega$ defined in \ref{['def:Omega_Phi']}.

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Example 1.4
  • Corollary 1.5
  • Example 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 19 more