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

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.