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Negative Eigenvalue Estimates for the 1D Schrödinger Operator with Measure-Potential.

Robert Fulsche1, Medet Nursultanov2,3, Grigori Rozenblum4

  • 1Institut für Analysis, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany.

Annales Henri Poincare
|February 20, 2026
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Summary
This summary is machine-generated.

This study analyzes the negative spectrum of a quantum operator with a measure potential. Researchers derived new eigenvalue estimates using Otelbaev

Keywords:
34L15Primary 47A75Secondary 34E15

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Area of Science:

  • Mathematical Physics
  • Spectral Theory
  • Quantum Mechanics

Background:

  • The study focuses on the spectral properties of differential operators.
  • Investigating operators with measure potentials is crucial in quantum mechanics.

Purpose of the Study:

  • To analyze the negative part of the spectrum for the operator -∂² - μ.
  • To derive estimates for eigenvalue counting functions and individual eigenvalues.
  • To establish Lieb-Thirring type estimates for the operator.

Main Methods:

  • The analysis involves the operator -∂² - μ defined on L²(ℝ).
  • A locally finite Radon measure μ ≥ 0 is used as the potential.
  • Otelbaev's function, an average of the measure potential, is a key tool.

Main Results:

  • Estimates for the eigenvalue counting function were obtained.
  • Estimates for individual eigenvalues were derived.
  • Lieb-Thirring type estimates were established.

Conclusions:

  • The research provides significant insights into the spectral properties of operators with measure potentials.
  • Otelbaev's function proves instrumental in obtaining these spectral estimates.
  • The findings contribute to the understanding of quantum mechanical systems with singular potentials.