Annales Henri Poincaré

, Volume 20, Issue 5, pp 1543–1582 | Cite as

Resonant Rigidity for Schrödinger Operators in Even Dimensions

  • T. J. ChristiansenEmail author


This paper studies the resonances of Schrödinger operators with bounded, compactly supported, real-valued potentials on \({{\mathbb {R}}}^d\), where the dimension d is even. If the potential V is non-trivial and \(d\not =4\), 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 \(-\varDelta +V_1\) and \(-\varDelta +V_2\) have the same poles, \(V_1,\;V_2\in L^\infty _c({{\mathbb {R}}}^d;{{\mathbb {R}}})\), \(k\in {{\mathbb {N}}}\), and if \(V_1\in H^k({{\mathbb {R}}}^d;{{\mathbb {R}}})\), then \(V_2\in H^k\) 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 \(L^\infty _c({{\mathbb {R}}}^d;{{\mathbb {R}}})\) cannot differ by a nonzero finite number of elements away from 0.


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It is a pleasure to thank Rafael Benguria, Peter Hislop, Antônio Sá Barreto, and Maciej Zworski for helpful conversations. The author is grateful to the Simons Foundation for its support through the Collaboration Grants for Mathematicians program.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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