Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case
Grigori RozenblumChalmers University of Technology, Göteborg, Sweden; St. Petersburg State University; Russian Federation
Eugene ShargorodskyKing's College London, UK; Technische Universität Dresden, Germany
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We consider self-adjoint operators of the form in a domain , where is an order pseudodifferential operator in and is a signed Borel measure with compact support in . Measure may contain singular component. For a wide class of measures we establish eigenvalue estimates for operator . In case of measure being absolutely continuous with respect to the Hausdorff measure on a Lipschitz surface of an arbitrary dimension, we find the eigenvalue asymptotics. The order of eigenvalue estimates and asymptotics does not depend on dimensional characteristics of the measure, in particular, on the dimension of the surface supporting the measure.