Resonant Rigidity for Schrödinger Operators in Even Dimensions

• Published in: Annales Henri Poincaré 2019-05, Vol.20 (5), p.1543-1582, Article 1543
• Main Author: Christiansen, T. J.
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Dynamical Systems and Ergodic Theory; Elementary Particles; Lower bounds; Mathematical and Computational Physics; Mathematical Methods in Physics; Operators (mathematics); Physics; Physics and Astronomy; Poles; Quantum Field Theory; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-019-00791-6

This paper studies the resonances of Schrödinger operators with bounded, compactly supported, real-valued potentials on R d , where the dimension d is even. If the potential V is non-trivial and d ≠ 4 , then the meromorphic continuation of the resolvent of the Schrödinger operator has infinitely many poles, with a quantitative lower bound on their density. A somewhat weaker statement holds if d = 4 . We prove several inverse-type results. If the meromorphic continuations of the resolvents of two Schrödinger operators - Δ + V 1 and - Δ + V 2 have the same poles, V 1 , V 2 ∈ L c ∞ ( R d ; R ) , k ∈ N , and if V 1 ∈ H k ( R d ; R ) , then V 2 ∈ H k as well. Moreover, we prove that certain sets of isoresonant potentials are compact. We also show that the poles of the resolvent for a smooth potential determine the heat coefficients and that the (resolvent) resonance sets of two potentials in L c ∞ ( R d ; R ) cannot differ by a nonzero finite number of elements away from 0.