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Resonances sets of Schrödinger operators

Yurii Belov, Pavel Gubkin

TL;DR

This work characterizes which resonance sets of one-dimensional Schrödinger operators on the half-line can be realized by compactly supported potentials. It proves that any Blaschke set $\Lambda$ contained in the lower-angle region $\{z:-\mathrm{Im}z\ge C|\mathrm{Re}z|\}$ can be embedded as resonances for some $q$ with compact support, and it provides sufficient conditions for realizability in wider domains via two complementary constructions: Hayman-type in Paley–Wiener spaces for the angle and Luxemburg–Korevaar constructions for broader regions. A detailed interpolation framework is developed, relying on $g\in\mathcal{PW}_\sigma$ and $H\in\mathcal{PW}_\gamma$ with prescribed zeros, together with contour/residue methods to guarantee convergence and the proper Jost-function structure. The results are shown to be sharp by constructing counterexamples that reveal intrinsic limits on how large a resonance set can be realized by any compactly supported potential, highlighting the delicate balance between Blaschke conditions, domain width, and resonance distribution.

Abstract

We prove that resonances of the Schrödinger operator with compactly supported potential can contain arbitrary subset of the angle $\{z: -\text{Im} z > C |\text{Re} z|\}$ that satisfies Blaschke condition. We also establish sufficient conditions for the subsets of wider domains.

Resonances sets of Schrödinger operators

TL;DR

This work characterizes which resonance sets of one-dimensional Schrödinger operators on the half-line can be realized by compactly supported potentials. It proves that any Blaschke set contained in the lower-angle region can be embedded as resonances for some with compact support, and it provides sufficient conditions for realizability in wider domains via two complementary constructions: Hayman-type in Paley–Wiener spaces for the angle and Luxemburg–Korevaar constructions for broader regions. A detailed interpolation framework is developed, relying on and with prescribed zeros, together with contour/residue methods to guarantee convergence and the proper Jost-function structure. The results are shown to be sharp by constructing counterexamples that reveal intrinsic limits on how large a resonance set can be realized by any compactly supported potential, highlighting the delicate balance between Blaschke conditions, domain width, and resonance distribution.

Abstract

We prove that resonances of the Schrödinger operator with compactly supported potential can contain arbitrary subset of the angle that satisfies Blaschke condition. We also establish sufficient conditions for the subsets of wider domains.
Paper Structure (10 sections, 11 theorems, 95 equations, 5 figures)

This paper contains 10 sections, 11 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Characterization of Jost functions
  3. Ideas of the proofs
  4. Points in the angle. Proof of Theorem \ref{['thm: angle']}
  5. Points in the wider domain. Proof of Theorem \ref{['thm: wider domain']}
  6. Luxemburg-Korevaar Construction
  7. Proof of Theorem \ref{['thm: H lower bound']}
  8. Points Clustering
  9. Proof of Theorem \ref{['thm: wider domain']}
  10. Sharpness of the results. Proof of Theorem \ref{['thm non res example']}

Key Result

Theorem 1.1

If $\Lambda$ satisfies the classical Blaschke condition $\sum_{\lambda\in \Lambda}\frac{|\mathop{\rm Im}\lambda|}{1 + |\lambda|^2} < \infty$ and for some $C > 0$ we have $\Lambda\subset\{z: -\mathop{\rm Im} z \geq C |\mathop{\rm Re} z|\}$ then there exists $q\in L^1(\mathbb R_+)$ with compact suppor

Figures (5)

  • Figure 1: The set $\Lambda$ is partitioned into the disjoint union $\Lambda = \cup_{n\geqslant 1}\Lambda_n$ and the contour $\Gamma_n$ is drawn around each of these sets.
  • Figure 2: The boundary of the angle $K_A$ and segments $K_A\cap \{\mathop{\rm Im} z = -h_n\}$ do not intersect the circles (drawn in red), where \ref{['eq: H(z) asymptotics']} may fail.
  • Figure 3: The contour $\Gamma_n$ surrounds the polygon $Q_n$ for every $n\geqslant 0$.
  • Figure 4: The set $U_{even}$ is the union of disjoint neighborhoods of $Q_{2n}$ because of the assertion $h_{n} + 2 \leqslant h_{n + 1}$.
  • Figure 5: Clusters visualization

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: E. Korotyaev, Theorem 1.1, korotyaev2004inverse
  • Corollary 2.2
  • proof
  • Theorem 5.1: Luxemburg, Korevaar, Theorem 5.2 luxemburg1971entire
  • Theorem 5.2
  • Lemma 5.3
  • proof
  • ...and 8 more