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Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schrödinger Operators

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Operator Algebras, Operator Theory and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 181))

Abstract

The aim of this paper is to study relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on L p(ℝn) perturbed by singular potentials.

Our approach to this problem is via the limit operators method. This method associates with each band-dominated operator A a family op(A) of so-called limit operators which reflect the properties of A at infinity. Consider the compactification of ℝn by the “infinitely distant” sphere S n−1. Then the set op(A) can be written as the union of its components op ηω (A) where ω runs through the points of S n−1 and where op ηω (A) collects all limit operators of A which reflect the properties of A if one tends to infinity “in the direction of ω”. Set \( sp_{n_\omega } A: = \cup _{A_h \in op_{\eta \omega } (A)} spA_h \).

We show that the distance of an eigenvalue λ ∉ sp ess A to sp ηω A determines the exponential decay of the λ-eigenfunctions of A in the direction of ω. We apply these results to estimate the exponential decay of eigenfunctions of electro-magnetic Schrödinger operators for a large class of electric potentials, in particular, for multiparticle Schrödinger operators and periodic Schrödinger operators perturbed by slowly oscillating at infinity potentials.

This work has been partially supported by CONACYT project 43432 and DFG Grant 444 MEX-112/2/05.

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Authors and Affiliations

  1. ESIME-Zacatenco, Instituto Politechnico National, Ed.1, 2-do piso Av. IPN, Mexico, D.F., 07738, Mexico

    Vladimir S. Rabinovich

  2. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289, Darmstadt, Germany

    Steffen Roch

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  1. Vladimir S. Rabinovich
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  2. Steffen Roch
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Editors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, U.T.L., Avenida Rovisco Pais, 1049-001, Lisboa, Portugal

    Maria Amélia Bastos , Amarino Brites Lebre  & Frank-Olme Speck ,  & 

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Israel Gohberg

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Rabinovich, V.S., Roch, S. (2008). Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schrödinger Operators. In: Bastos, M.A., Lebre, A.B., Speck, FO., Gohberg, I. (eds) Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol 181. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8684-9_18

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