Investigations of the numerical range of a operator matrix. (Russian. English summary) Zbl 1413.81022
Summary: We consider a \(2\times2\) operator matrix \(A\) (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on \({\mathrm d}\)-dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range \(W(A)\) of this operator in detail by terms of its matrix entries for all dimensions of the torus \({\mathbf T}^{\mathrm d}\). Moreover, we study the cases when the set \(W(A)\) is closed and give necessary and sufficient conditions under which the spectrum of \(A\) coincides with its numerical range.
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 47N50 | Applications of operator theory in the physical sciences |
Keywords:
operator matrix; generalized Friedrichs model; Fock space; numerical range; point and approximate point spectra; annihilation and creation operators; first Schur complimentReferences:
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