Investigations in the Spectral Properties of Operators with Distant Perturbations (survey)

We propose a chronological overview of researches on operators with distant perturbations. Let us explain what "distant perturbations" mean. An elementary example of the operator with distant perturbations is a differential operator of the second order with two finite potentials. Supports of these operators are at a great distance from each other, i.e. they are \distant".The study of such operators has been performed since the middle of the last century, mostly by foreign researchers see eg. R. Ahlrichs, T. Aktosun, M. Klaus, P. Aventini, P. Exner, E.B. Davies, V. Graffi, E.V. Harrell II, H.J. Silverstone, M. Mebkhout, R. Hoegh-Krohn, W. Hun ziker, V. Kostrykin, R. Schrader, J.D. Morgan (III), Y. Pinchover, O.K. Reity, H. Tamura, X. Wang, Y. Wang, S. Kondej, B. Simon, I. Veselic, D.I. Borisov, A.M. Golovina). The main objects of their investigation were the asymptotic behaviors of eigenvalues and corresponding eigenfunctions of perturbed operators. In several papers the research was focused on resolvents and eigenvalues of perturbed operator arising from the edge of the essential spectrum. The main results of the past century are the first members of the asymptotics of perturbed eigenvalues and the corresponding eigenfunctions and the first members of the asymptotics of resolvents of the perturbed operators. The main results of the last fifteen years are full asymptotic expansions for the eigenvalues and their corresponding functions and an explicit formula for the resolvent of the perturbed operator.In this paper, we also note that up to 2004 only different kind of potentials were considered as perturbing operators, and Laplace and Dirac operators were considered as unperturbed operators. Only since 2004, nonpotential perturbing operators appeared in the literature. Since 2012, an arbitrary elliptic differential operator is considered as an unperturbed operator.We propose a classification of investigations on distant perturbations, based on the spectral properties of the operators:1) investigations into the eigenvalues and the corresponding eigenfunctions of the Laplaceoperator with distant potentials;(a) in the case of a simple limit eigenvalue;(b) in the case of a multiple limit eigenvalue;2) investigations into the resolvent of the Laplace operator with several distant potentials;3) investigations into asymptotic behavior of the eigenvalues arising from the edge of the essential spectrum of the unperturbed operator.In conclusion, we formulate open problems in this theory:1. What kind of behavior of the eigenvalues and the corresponding eigenfunctions arising from the edge of the essential spectrum? Under what conditions do they arise? What is their asymptotic expansion?2. What are the first members of perturbed eigenvalues in the case of an arbitrary finite number of distant perturbations?DOI: 10.7463/mathm.0215.077685