Limiting Absorption Principle for Discrete Schrödinger Operators with a Wigner–von Neumann Potential and a Slowly Decaying Potential

• Published in: Annales Henri Poincaré Vol. 22; no. 1; pp. 83 - 120
• Main Authors: Golénia, Sylvain, Mandich, Marc-Adrien
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.01.2021; Springer Nature B.V; Springer Verlag
• Subjects: Absorption; Classical and Quantum Gravitation; Constraining; Decay rate; Dynamical Systems and Ergodic Theory; Elementary Particles; Functional Analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical Physics; Mathematics; Operators (mathematics); Original Paper; Perturbation; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Spectral Theory; Theoretical; 81Q10; 47A10; 47B25; 39A70; 2010 Mathematics Subject Classification. 39A70; 47A10 limiting absorption principle; discrete Schrödinger operator; Wigner-von Neumann potential; polylogarithms; Loewner's theorem; weighted Mourre theory; Mourre theory
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-020-00971-9

We consider discrete Schrödinger operators on Z d for which the perturbation consists of the sum of a long-range-type potential and a Wigner–von Neumann-type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in Mandich (J Funct Anal 272(6):2235–2272, 2017). To our knowledge, this is a new result even in the one-dimensional case. The improvement is twofold. It weakens the assumptions on the long-range potential and provides better LAP weights. Both upgrades include logarithmic terms. The proof relies on the fact that some particular functions that contain logarithmic terms, are operator monotone. This fact is proved using Loewner’s theorem and Nevanlinna functions.