Spectral Properties of Schrödinger Operators with Decaying Potentials

2005/09/28 by Sergey A. Denisov, Denisov, Sergey A., Alexander Kiselev +1 · 2 citations
Computer Science · Mathematics · #34L40 #35Q40 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP #msc:34L40 #msc:35Q40

paper · pdf · doi:10.48550/arxiv.math/0509668

25 pages

arxiv created 2005/09/28 · openalex publication_date 2005/09/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We review recent advances in the spectral theory of Schrödinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994 International Congress on Mathematical Physics in Paris. The one-dimensional picture is now fairly complete, and provides many striking spectral examples. The multidimensional picture is still far from clear and may require deep original ideas for further progress. It might hold the keys for better understanding of a wide range of spectral and dynamical phenomena for Schrödinger operators in higher dimensions.

Cited by

• Spatial asymptotics of Green's function for elliptic operators and\n applications: a.c. spectral type, wave operators for wave equations
• Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under exponentially decaying perturbations

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