Threshold phenomenon for a family of the generalized Friedrichs models with the perturbation of rank one

S. N. Lakaev ; M. Darus ; S. T. Dustov

Ufa mathematical journal, Tome 11 (2019) no. 4, pp. 140-150

Résumé

In this work we consider a family $H_\mu(p),$ $\mu>0,$ $p\in\mathbb{T}^3$, of the generalized Friedrichs models with the perturbation of rank one. This system describes a system of two particles moving on the three dimensional lattice $\mathbb{Z}^3$ and interacting via a pair of local repulsive potentials. One of the reasons to consider such family of the generalized Friedrichs models is that this family generalizes and involves some important behaviors of the Hamiltonians for systems of both bosons and fermions on lattices. In the work, we study the existence or absence of the eigenvalues of the operator $H_\mu(p)$ located outside the essential spectrum depending on the values of $\mu>0$ and $p\in U_{\delta}(p_{\,0})\subset\mathbb{T}^3$. We prove a analytic dependence on the parameters for such eigenvalue and an associated eigenfunction and the latter is found in a certain explicit form. We prove the existence of coupling constant threshold $\mu=\mu(p)>0$ for the operator $H_\mu(p)$, $p\in U_{\delta}(p_{\,0})$, namely, we show that the operator $H_\mu(p)$ has no eigenvalue for all $0\mu\mu(p)$ and there exists a unique eigenvalue $z(\mu,p)$ for each $\mu>\mu(p)$ and this eigenvalue is located above the threshold $z=M(p)$. We find necessary and sufficient conditions allowing us to determine whether the threshold $z=M(p)$ is an eigenvalue or a virtual level or a regular point in the essential spectrum of the operator $H_\mu(p),$ $p\in U_{\delta}(p_{\,0})$.

Détail

Keywords: coupling constant threshold, repulsive potential, eigenvalue, generalized Friedrichs model, regular point.

Bibliographie

[1] S. N. Lakaev, A. Ibrahim, Sh. H. Kurbanov, “Threshold effects for the generalized Friedrichs model with the perturbation of rank one”, Abstr. Appl. Anal., 2012 (2012), 180953 | DOI | MR | Zbl

[2] Funct. Anal. Appl., 17:4 (1983), 317–319 | DOI | MR | Zbl

[3] J. Soviet Math., 45:6 (1989), 1540–1565 | DOI | MR | Zbl

[4] Funct. Anal. Appl., 38:3 (2004), 202–216 | DOI | DOI | MR | Zbl

[5] J. Hecker Denschlag, A. J. Daley, “Exotic atom pairs: Repulsively bound states in an optical lattice”, Proceedings of the International School of Physics “Enrico Fermi”, 164, 2007, 677–696

[6] K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P. Bc̈hler, P. Zoller, “Repulsively bound atom pairs in an optical lattice”, Nature, 441 (2006), 853–856 | DOI

[7] Theor. Math. Phys., 103:1 (1995), 390–397 | DOI | MR

[8] A. Mogilner, “Hamiltonians in solid state physics as multi-particle discrete Schrödinger operators: Problems and results”, Adv. Soviet Math., 5 (1991), 139–194 | MR | Zbl

[9] M. Reed, B. Simon, Methods of modern mathematical physics, v. III, Scattering theory, Academic Press, New York, 1979 | MR | Zbl

[10] K. O. Friedrichs, “On the perturbation of continuous spectra”, Comm. Appl. Math., 1:4 (1948), 361–406 | DOI | MR | Zbl

[11] K. O. Friedrichs, Perturbation of spectra in Hilbert space, Amer. Math. Soc., Providence, RI, 2008 | MR

[12] V. A. Malishev, R. A. Minlos, Linear infinite-particle operators, Transl. Math. Monogr., 143, Amer. Math. Soc., Providence, RI, 1995 | DOI | MR

[13] S. Albeverio, S. N. Lakaev, Z. I. Muminov, “The threshold effects for a family of Generalized Friedrichs models under rank one perturbations”, J. Math. Anal. Appl., 330:2 (2007), 1152–1168 | DOI | MR | Zbl

[14] S. N. Lakaev, M. Darus, Sh. H. Kurbanov, “Puiseux series expansion for an eigenvalue of the generalized Friedrichs model with perturbation of rank 1”, J. Phys. A: Math. Theor., 46:20 (2013), 205304 | DOI | MR | Zbl

[15] M. Klaus, B. Simon, “Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case”, Ann. Phys., 130:2 (1980), 251–281 | DOI | MR | Zbl

[16] S. N. Lakaev, Sh. Yu. Holmatov, “Asymptotics of Eigenvalues of a two-particle Schrödinger operators on lattices with zero range interaction”, J. Phys. A: Math. Theor., 44:13 (2011), 135304 | DOI | MR | Zbl

[17] S. Albeverio, S. N. Lakaev, K. A. Makarov, Z. I. Muminov, “The threshold effects for the two-particle Hamiltonians on lattices”, Comm. Math. Phys., 262:1 (2006), 91–115 | DOI | MR | Zbl

[18] M. Reed, B. Simon, Methods of modern mathematical physics, v. IV, Analysis of operators, Academic Press, New York, 1978 | MR | Zbl

[19] R. Courant, D. Hilbert, Methods of mathematical physics, v. 2, Partial differential equations, John Wiley Sons, New York, 1966 | MR

[20] Theor. Math. Phys., 91:1 (1992), 362–372 | DOI | MR | MR

[21] M. V. Fedoryuk, Asymptotics of integrals and series, Nauka, M., 1987 (in Russian) | MR