Eigenvalues of the Birman-Schwinger operator for singular measures: the noncritical case

2021/07/09 by Grigori Rozenblum, Rozenblum, Grigori, Grigory Tashchiyan +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2107.04682

Abstract

In a domain Ω⊆ ℝ^N we consider compact, Birman-Schwinger type, operators of the form T_P,\mathfrakA=\mathfrakA^*P\mathfrakA; here P is a singular Borel measure in Ω and \mathfrakA is a noncritical order -l≠ -N/2 pseudodifferential operator. For a class of such operators, we obtain estimates and a proper version of H.Weyl's asymptotic law for eigenvalues, with order depending on dimensional characteristics of the measure. A version of the CLR estimate for singular measures is proved. For non-selfadjoint operators of the form P₂ \mathfrakA P₁ and \mathfrakA₂ P \mathfrakA₁ with singular measures P,P₁,P₂ and negative order pseudodifferential operators \mathfrakA,\mathfrakA₁,\mathfrakA₂ we obtain estimates for singular numbers.

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• Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures

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