Spectral properties of the Hill operator

Abstract

Let L 2 [0, ω] be the Hilbert space of complex measurable (classes of) functions whose modulus is square-summable with inner product of the formWe consider the Hill differential operatorIt is assumed that the potential v belongs to the Hilbert space L 2 [0, ω] and its Fourier series expansionis used everywhere below. The operator L θ , θ ∈ [0, 1], can be expressed asThe operator V is the operator of multiplication by the potential v with domainThe operator L 0 θ is a self-adjoint nonnegative operator with compact resolvent.The spectrum σ(L 0 θ ) of the operator L 0 θ is described as follows:(a) for the case θ ∈ {0, 1}, the eigenvalues are of the form λ n = λ n,θ = π(2n + θ) ω 2 ,

The matrix analysis of spectral projections for the perturbed self-adjoint operators

“…Отдельно отметим работу [33], в которой разработана новая модификация метода подобных операторов, предназначенная для исследования оператора L bc и с ее помощью проведено полное исследование спектральных свойств оператора L bc . Результаты работы [33] анонсировались в [34]. В этих работах вопросы анализа матрицы спектральных проекторов и резольвенты не рассматривались.…”

“…Отметим, что кроме упомянутых работ [29] - [34] операторы L bc исследуются также в работах других авторов (см., например, [35] - [37]). В работе [35] приведена асимптотика собственных значений оператора L per , а в [37] рассматривалась асимптотика спектра негладких возмущенных дифференциальных операторов 2m-ого порядка.…”

Method of similar operators in the problem of bi-invariant subspaces

“…It is noteworthy that in narrower contexts, such as that of Hilbert spaces, modifications of the method of similar operators can yield more precise outcomes (as demonstrated, for example, in [22,30,32,33]). In other specialized settings, different perturbation theory methods for analyzing the spectral properties of linear operators may offer results of comparable or heightened quality [60,72,73,74,75,79,80,84].…”

On the Riesz basis property of the root functions of a discontinuous boundary problem

“…20 For real potential, the several spectral properties were investigated in Marchenko, 21 Djakov and Mityagin, 22 Baranets'kyi et al 23 The spectral character of the Hill operator studied in Baskakov. 24,25 These papers also provide wide sources (eg, see other works 26,27 ). We must also note that Riesz basisness of problem (1) with strongly regular boundary conditions and conditions (6) and (7) was studied in Muravei.…”

“…A classification on the conditions for Sturm Liouville operator under which root functions form a Riesz basis in L 2 (0,1) is established in Makin . For real potential, the several spectral properties were investigated in Marchenko, Djakov and Mityagin, Baranets'kyi et al The spectral character of the Hill operator studied in Baskakov . These papers also provide wide sources (eg, see other works).…”