The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential. (English. Russian original) Zbl 1441.47054

Baskakov, Anatoly G.; Polyakov, Dmitry M.

Sb. Math. 208, No. 1, 1-43 (2017); translation from Mat. Sb. 208, No. 1, 3-47 (2017).

Let \(L_\theta\) be the operator generated by the expression \(-x'' - v x\) on a finite interval \([0, \omega]\) together with the quasiperiodic boundary conditions \[ x(\omega) = e^{i \pi \theta} x(0), \qquad x'(\omega) = e^{i \pi \theta} x'(0), \] where the potential \(v\) is a complex-valued square summable function and \(\theta \in [0, 1]\). The authors first prove that \(L_\theta\) is similar to an integro-differential operator of particular form. They then apply this result to obtain refined asymptotic estimates for the eigenvalues in terms of the Fourier coefficients of \(v\), estimates for the equiconvergence of spectral decompositions for this operator and the corresponding free operator (i.e., with \(v \equiv 0\)), and asymptotic formulas for the semigroup of operators generated by \(-L_\theta\). They also obtain sufficient conditions for \(L_\theta\) to be spectral in the sense of Dunford. Some of these results were previously announced without proof in [A. G. Baskakov and D. M. Polyakov, Math. Notes 99, No. 4, 598–602 (2016; Zbl 1371.34129)].

Reviewer: Namig Guliyev (Baku)

MSC:

47E05	General theory of ordinary differential operators
34L20	Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47G20	Integro-differential operators
47D03	Groups and semigroups of linear operators

Keywords:

Hill’s differential operator; method of similar operators; spectrum of an operator; asymptotic behaviour of eigenvalues; semigroup of operators; spectral operators

Citations:

Zbl 1371.34129