×

On the finiteness of the discrete spectrum of a \(3\times 3\) operator matrix. (English) Zbl 1363.81012

The author considers an operator matrix \(H\) associated with the lattice system describing three particles in interaction without conservation of the number of particles; this is a lattice analog of the spin-boson Hamiltonian (see [R. Minlos and H. Spohn, in: Topics in statistical and theoretical physics. F. A. Berezin memorial volume. Transl. ed. by A. B. Sossinsky. Providence, RI: American Mathematical Society. 159–193 (1996; Zbl 0881.47049)]. The main results deal with the structure of the essential and point spectra of \(H\). In particular, the author finds conditions guaranteeing the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P15 Estimates of eigenvalues in context of PDEs
47N50 Applications of operator theory in the physical sciences

Citations:

Zbl 0881.47049
PDFBibTeX XMLCite