Spectral theory of Schrödinger operators with potentials that are measures supported on \Bbb N

2025/09/24 by Safronov, Oleg
#34A37 #47A10 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2509.20320

Abstract

We discuss spectral properties of the one-dimensional Schrödinger operator with a potential of the form ∑ V(n)δ(x-n). Our main result says that the absolutely continuous spectum of such an operator covers an interval [α²,β²], if V∈ ℓ⁴ and the Fourier series ∑ e^{2i kn}V(n) is a function of k that is square integrable over [α,β]. We prove that this result is sharp by constructing examples of potentials V∉ℓ² for which the spectrum of the Schrödinger operator is singular.

Citations

• Lieb-Thirring Inequalities for Jacobi Matrices
• Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory
• Absolutely continuous spectrum for one-dimensional Schr "odinger\n operators with slowly decaying potentials: some optimal results

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