Conditions for the existence of eigenvalues of a three-particle lattice model Hamiltonian. (English. Russian original) Zbl 1541.81063
Russ. Math. 67, No. 7, 1-8 (2023); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2023, No. 7, 3-12 (2023).
Summary: In this article, we present a three-particle lattice model Hamiltonian \({{H}_{{\mu ,\lambda }}}\), \(\mu ,\lambda > 0\) by making use nonlocal potential. The Hamiltonian under consideration acts as a tensor sum of two Friedrichs models \({{h}_{{\mu ,\lambda }}}\) which comprises a rank 2 perturbation associated with a system of three quantum particles on a \(d\)-dimensional lattice. The current study investigates the number of eigenvalues associated with the Hamiltonian. Furthermore, we provide the suitable conditions on the existence of eigenvalues localized inside, in the gap and below the bottom of the essential spectrum of \({{H}_{{\mu ,\lambda }}}\).
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 70B05 | Kinematics of a particle |
| 70F07 | Three-body problems |
| 70H05 | Hamilton’s equations |
| 35P05 | General topics in linear spectral theory for PDEs |
| 35B20 | Perturbations in context of PDEs |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 47A10 | Spectrum, resolvent |
Keywords:
model Hamiltonian; lattice; perturbation; nonlocal potential; tensor sum; Friedrichs model; spectrumReferences:
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