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Analysis of the discrete Spectrum of the family SPECTRUM of \(3\times 3\) operator matrices. (English) Zbl 1453.81029

Summary: We consider the family of \(3\times 3\) operator matrices \(\mathbf{H}(K)\), \(K\in\mathbb{T}^3:=(-\pi; \pi]^3\) associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set \(\Lambda\subset\mathbb{T}^3\) to prove the existence of infinitely many eigenvalues of \(\mathbf{H}(K)\) for all \(K\in\Lambda\) when the associated Friedrichs model has a zero energy resonance. It is found that for every \(K\in\Lambda\), the number \(N(K, z)\) of eigenvalues of \(\mathbf{H}(K)\) lying on the left of \(z, z<0\), satisfies the asymptotic relation \(\lim\limits_{z\to -0}N(K,z)|\log|z||^{-1}=\mathcal{U}_0\) with \(0<\mathcal{U}_0<\infty\), independently on the cardinality of \(\Lambda\). Moreover, we prove that for any \(K\in\Lambda\) the operator \(\mathbf{H}(K)\) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.

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
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