The discrete spectrum in the singular Friedrichs model

1998/06/15 by D. R. Yafaev, D. Yafaev, Yafaev, D.
Mathematics · Physics and Astronomy · #35J10 #47A75 #81U20 #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Quantum optics and atomic interactions #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:35J10 #msc:47A75 #msc:81U20

paper · pdf · doi:10.48550/arxiv.math-ph/9806009

Latex

arxiv created 1998/06/15 · openalex publication_date 1998/06/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A typical result of the paper is the following. Let H_γ=H₀ +γV where H₀ is multiplication by |x|^2l and V is an integral operator with kernel cos< x,y\rang le in the space L₂(R^d). If l=d/2+ 2k for some k= 0,1,..., then the operator H_γ has infinite number of negative eigenvalues for any coupling constant γ≠ 0. For other values of l, the negative spectrum of H_γ is infinite for |γ|> σ_l where σ_l is some explicit positive constant. In the case ± γ∈ (0,σ_l], the number N^(±)_l of negative eigenvalues of H_γ is finite and does not depend on γ. We calculate N^(±)_l.

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