×

On the discrete spectrum of a model operator in fermionic Fock space. (English) Zbl 1433.81086

Summary: We consider a model operator \(H\) associated with a system describing three particles in interaction, without conservation of the number of particles. The operator \(H\) acts in the direct sum of zero-, one-, and two-particle subspaces of the fermionic Fock space \(\mathcal{F}_a(L^2(\mathbb{T}^3))\) over \(L^2(\mathbb{T}^3)\). We admit a general form for the “kinetic” part of the Hamiltonian \(H\), which contains a parameter \(\gamma\) to distinguish the two identical particles from the third one. (i) We find a critical value \(\gamma^{*}\) for the parameter \(\gamma\) that allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only for \(\gamma < \gamma^{*}\) the Efimov effect is absent, while this effect exists for any \(\gamma > \gamma^{*}\). (ii) In the case \(\gamma > \gamma^{*}\) , we also establish the following asymptotics for the number \(N(z)\) of eigenvalues of \(H\) below \(z < E_{\min} = \inf \sigma_{\mathrm{ess}} \left(H\right) : \lim_{z \rightarrow E_{\min}} \left(N \left(z\right) / \left|\log \left|E_{\min} - z\right|\right|\right) =\mathcal{U}_0 \left(\gamma\right) \left(\mathcal{U}_0 \left(\gamma\right) > 0\right)\), for all \(\gamma > \gamma^*\).

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47A10 Spectrum, resolvent
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Efimov, V., Energy levels of three resonantly interacting particles, Nuclear Physics A, 210, 157-158 (1973) · doi:10.1016/0375-9474(73)90510-1
[2] Albeverio, S.; Høegh-Krohn, R.; Wu, T. T., A class of exactly solvable three-body quantum mechanical problems and the universal low energy behavior, Physics Letters A, 83, 3, 105-109 (1981) · doi:10.1016/0375-9601(81)90507-7
[3] Albeverio, S.; Lakaev, S. N.; Makarov, K. A., The Efimov effect and an extended Szegő-Kac limit theorem, Letters in Mathematical Physics, 43, 1, 73-85 (1998) · Zbl 0903.45003 · doi:10.1023/A:1007466105600
[4] Amado, R. D.; Noble, J. V., On efimov’s effect: a new pathology of three-particle systems, Physics Letters B, 35, 1, 25-27 (1971)
[5] Dell’Antonio, G. F.; Figari, R.; Teta, A., Hamiltonians for systems of \(N\) particles interacting through point interactions, Annales de l’Institut Henri Poincaré, 60, 3, 253-290 (1994) · Zbl 0808.35113
[6] Faddeev, L. D.; Merkuriev, S. P., Quantum Scattering Theory for Several Particle Systems (1993), New York, NY, USA: Kluwer Academic, New York, NY, USA · Zbl 0797.47005
[7] Ovchinnikov, Yu. N.; Sigal, I. M., Number of bound states of three-body systems and Efimov’s effect, Annals of Physics, 123, 2, 274-295 (1979) · doi:10.1016/0003-4916(79)90339-7
[8] Sobolev, A. V., The Efimov effect. Discrete spectrum asymptotics, Communications in Mathematical Physics, 156, 1, 101-126 (1993) · Zbl 0785.35070 · doi:10.1007/BF02096734
[9] Tamura, H., The Efimov effect of three-body Schrödinger operators, Journal of Functional Analysis, 95, 2, 433-459 (1991) · Zbl 0761.35078 · doi:10.1016/0022-1236(91)90038-7
[10] Tamura, H., Asymptotics for the number of negative eigenvalues of three-body Schrödinger operators with Efimov effect, Spectral and Scattering Theory and Applications. Spectral and Scattering Theory and Applications, Advanced Studies in Pure Mathematics, 23, 311-322 (1994), Tokyo, Japan: Mathematical Society of Japan, Tokyo, Japan · Zbl 0836.35110
[11] Jafaev, D. R., On the theory of the discrete spectrum of the three-particle schrِdinger operator, Mathematics of the USSR-Sbornik, 23, 4, 535-559 (1974) · Zbl 0342.35041 · doi:10.1070/SM1974v023n04ABEH001730
[12] Wang, X. P., On the existence of the \(N\)-body Efimov effect, Journal of Functional Analysis, 209, 1, 137-161 (2004) · Zbl 1059.81061 · doi:10.1016/S0022-1236(03)00170-8
[13] Minlos, R. A.; Spohn, H., The three-body problem in radioactive decay: the case of one atom and at most two photons, Topics in Statistical and Theoretical Physics. Topics in Statistical and Theoretical Physics, Transactions of the American Mathematical Society Series 2, 177, 159-193 (1996), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0881.47049
[14] Malyshev, V. A.; Minlos, R. A., Linear Infinite-Particle Operators (Translations of Mathematical Monographs), 143 (1995), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0827.46065
[15] Mattis, D. C., The few-body problem on a lattice, Reviews of Modern Physics, 58, 2, 361-379 (1986) · doi:10.1103/RevModPhys.58.361
[16] Mogilner, A. I., Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: problems and results, Many-particle Hamiltonians: Spectra and Scattering. Many-particle Hamiltonians: Spectra and Scattering, Advances in Soviet Mathematics, 5, 139-194 (1991), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0741.34055
[17] Buhler, C.; Yunoki, S.; Moreo, A., Magnetic domains and stripes in a spin-fermion model for cuprates, Physical Review Letters, 84, 12, 2690-2693 (2000) · doi:10.1103/PhysRevLett.84.2690
[18] Friedrichs, K. O., Perturbation of Spectra in Hilbert Space (1965), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0142.11001
[19] Mel’nikov, A. M., On the lower branches of the spectrum of two fermions interacting with a boson gas (exiton), Russian Mathematical Surveys, 50, 4, 824-825 (1995) · Zbl 0887.46046 · doi:10.1070/RM1995v050n04ABEH002587
[20] Sigal, I. M.; Soffer, A.; Zielinski, L., On the spectral properties of Hamiltonians without conservation of the particle number, Journal of Mathematical Physics, 43, 4, 1844-1855 (2002) · Zbl 1059.81060 · doi:10.1063/1.1452302
[21] Zhukov, Yu. V.; Minlos, R. A., The spectrum and scattering in the “spin-boson” model with at most three photons, Theoretical and Mathematical Physics, 103, 1, 398-411 (1995) · Zbl 0863.47056 · doi:10.1007/BF02069784
[22] Albeverio, S.; Lakaev, S. N.; Rasulov, T. H., On the spectrum of an hamiltonian in fock space. Discrete spectrum asymptotics, Journal of Statistical Physics, 127, 2, 191-220 (2007) · Zbl 1126.81022 · doi:10.1007/s10955-006-9240-6
[23] Albeverio, S.; Lakaev, S. N.; Rasulov, T. H., The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles, Methods of Functional Analysis and Topology, 13, 1, 1-16 (2007) · Zbl 1113.81051
[24] Albeverio, S.; Lakaev, S. N.; Muminov, Z. I., Schrödinger operators on lattices. The Efimov effect and discrete spectrum asymptotics, Annales Henri Poincaré, 5, 4, 743-772 (2004) · Zbl 1056.81026 · doi:10.1007/s00023-004-0181-9
[25] Lakaev, S. N., The efimov’s effect of a system of three identical quantum iattice particles, Funktsional’nyi Analiz i Ego Prilozheniya, 27, 3, 15-28 (1993) · Zbl 0844.47041 · doi:10.1007/BF01087534
[26] Dell’Antonio, G. F.; Muminov, Z. I.; Shermatova, Y. M., On the number of eigenvalues of a model operator related to a system of three particles on lattices, Journal of Physics A, 44, 31 (2011) · Zbl 1225.81062 · doi:10.1088/1751-8113/44/31/315302
[27] Petrov, D. S., Three-body problem in Fermi gases with short-range interparticle interaction, Physical Review A, 67, 1 (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.