On necessary and sufficient condition in theory of regularized traces

Z. Yu. Fazullin   ; N. F. Abuzyarova

Ufa mathematical journal, Tome 12 (2020) no. 4, pp. 90-98

Résumé

The present work is devoted to studying the regularized trace formulae for symmetric $L_0$-compact perturbations of a self-adjoint lower semi-bounded operator $L_0$ with a discrete spectrum in a separable Hilbert space. By now, the studies of the regularized trace formulae for the perturbations of abstract self-adjoint discrete operators were mostly aimed on finding a sufficient condition, under which the regularized sum with brackets minus first or several leading terms of the perturbation theory vanished. This condition was formulated in terms of spectral characteristics of an unperturbed operator $L_0$ depending on the belonging of a perturbing operator $V$ to some class. In particular, recently, the traces formulae for model two-dimensional operators in mathematical physics have been intensively studied with a perturbation described by the multiplication operator. Here we study a necessary and sufficient condition for two cases, namely, as the regularized trace with brackets and deduction of the first corrector of the perturbation theory vanishes or is equal to a finite number. We consider a certain summation bracket, which usually arises in the theory of regularized traces of the perturbations of partial differential operators.

Détail

Keywords: trace of an operator, resolvent, perturbation theory, discrete spectrum.
Mots-clés : trace formula

Bibliographie

[1] I.M. Gelfand, B.M. Levitan, “On a simple identity for eigenvalues of second differential oeprator”, Doklady AN SSSR, 88 (1953), 593–596 (in Russian) | Zbl

[2] L.A. Dikii, “On a formula of Gelfand-Levitan”, Uspekhi Matem. Nauk, 8:2 119–123 (1953) (in Russian) | MR

[3] V.A. Sadovnichii, V.E. Podolskii, “Traces of operators”, Russ. Math. Surveys, 61:5 (2006), 885–953 | DOI | MR | Zbl

[4] V.A. Sadovnichij, V.V. Dubrovskij, V.A. Lyubishkin, “Traces of discrete operators”, Sov. Math. Dokl., 25 (1982), 759–761 | MR | Zbl

[5] V.A. Sadovnichii, V.E. Podolskii, “Traces of operators with relatively compact perturbations”, Sb. Math., 193:2 (2002), 279–302 | DOI | MR

[6] Kh.Kh. Murtazin, Z.Yu. Fazullin, “Non-nuclear perturbations of discrete operators and trace formulae”, Sb. Math., 196:12 (2005), 1841–1874 | DOI | MR | Zbl

[7] V.A. Sadovnichii, V. V. Dubrovskii, “A classical regularized trace formula for the eigenvalues of the Laplace-Beltrami operator with a potential on the sphere $S^2$”, Soviet Math. Dokl., 44:1 (1992), 56–58 | MR

[8] V.A. Sadovnichij, Z.Yu. Fazullin, “A formula for the first regularized trace of a perturbed Laplace-Beltrami operator”, Diff. Equat., 37:3 (2001), 430–438 | DOI | MR | Zbl

[9] Z.Yu. Fazullin, Kh.Kh. Murtazin, “Regularized trace of a two-dimensional harmonic oscillator”, Sb. Math., 192:5 (2001), 725–761 | DOI | MR | Zbl

[10] Kh.Kh. Murtazin, F.Yu. Fazullin, “The spectrum and trace formula for the two-dimensional Schrödinger operator in a homogeneous magnetic field”, Dokl. Math., 67:3 (2003), 426–428 | MR | Zbl

[11] E. Korotyaev, A. Pushniski, “A trace formula and high-energy spectral asymptotics for the perturbed Landau hamiltonian”, Func. Anal., 217:1 (2004), 221–248 | DOI | MR | Zbl

[12] A.I. Atnagulov, V.A. Sadovnichii, Z.Yu. Fazullin, “Properties of the resolvent of the Laplace operator on a two-dimensional sphere and a trace formula”, Ufa Math. J., 8:3 (2016), 22–40 | DOI | MR | Zbl

[13] Z.Yu. Fazullin, I.G. Nugaeva, “Spectrum and a trace formula for a compactly supported perturbation of the 2D harmonic oscillator in a strip”, Diff. Equat., 55:5 (2019), 677–687 | DOI | MR | Zbl