Spectral analysis of an integro-differential operator with a degenerate kernel

We consider operator ℒ acting in the Hilbert space 𝐿2[0, 1] defined by the integro-differential expression (ℒ𝑥)(𝑡) = -¨𝑥(𝑡) - ∫︀1 0 𝐾(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 with the domain 𝐷(ℒ) = {𝑥 ∈ 𝑊2 2 [0, 1], 𝑥(0) = 𝑥(1) = 0}, where 𝑊2 2 [0, 1] is the Sobolev space {𝑥 ∈ 𝐿2[0, 1] : 𝑥′ is absolutely continuous, 𝑥′′ ∈ 𝐿2[0, 1]}, and the boundary conditions 𝑥(0) = 𝑥(1) = 0. To study spectral properties of the operator ℒ, it is represented in the form (ℒ𝑥)(𝑡) = (𝐴𝑥)(𝑡) - (𝐵𝑥)(𝑡), where with (𝐴𝑥)(𝑡) = -¨𝑥(𝑡), (𝐵𝑥)(𝑡) = = ∫︀1 0 𝐾(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠. Operator is considered an unperturbed operator, is a perturbation, which is an integral operator with a degenerate kernel 𝐾(𝑡, 𝑠) = = Σ︀𝑘 𝑖=1 𝑝𝑖(𝑡)𝑞𝑖(𝑠), 𝑝𝑖, ∈ 𝐿2[0, 1]. As a method of studying spectral properties of the operator - the method of similar operators serves. One of the main results is Theorem 3. Let for any functions 𝑝𝑖, 𝑞𝑖, = 1, 𝑘, belonging to a Hilbert space 𝐿2[0, 1], and for the sequences 1, 2 : N → R+ = [0,∞), defined by formulas 1(𝑛) = 1 2 2 (︃ Σ︁ 𝑚≥1 𝑚̸=𝑛 𝑛2 (︃ sup Σ︀𝑘 𝑖=1 𝑞sin 𝑝sin 𝑗 )︃2 + 𝑚2 (︃ sup | Σ︀𝑘 𝑖=1 𝑞sin 𝑝sin | )︃2 |𝑛2 - 𝑚2|2 )︃1 2