Spectral properties of second order differential operator determined by non-local boundary conditions

In this work we study the spectral properties of the operator acting in the Hilbert space 𝐿2[0, 2 ] defined by the differential expression ℒ𝑦 = = -¨𝑦 + and nonlocal boundary conditions 𝑦(0) = 𝑦(2 ) + ∫︁2 0 𝑎0(𝑡)𝑦(𝑡)𝑑𝑡, 𝑦˙(0) = 𝑦˙(2 ) + ∫︁2 0 𝑎1(𝑡)𝑦(𝑡)𝑑𝑡. Here 𝑎0 and 𝑎1 are functions from 𝐿2[0, 2 ]. To investigate spectrum of the operator, ℒ is used adjoint of the operator ℒ* one defined by the differential expression (ℒ*𝑥)(𝑡) = (𝐴𝑥)(𝑡) - (𝐵𝑥)(𝑡) and boundary conditions 𝑥(0) = 𝑥(2 ), 𝑥˙ (0) = 𝑥˙ (2 ), with generated by the differential expression = -¨𝑥 + with the domain 𝐷(𝐴) = {𝑥 ∈ 𝐿2[0, 2 ] : 𝑥, 𝑥˙ ∈ 𝐶[0, 2 ], 𝑥¨ ∈ 𝐿2[0, 2 ], 𝑥(0) = 𝑥(2 ), 𝑥˙ (0) = 𝑥˙ (2 )}, and (𝐵𝑥)(𝑡) = 𝑥˙ (2 )𝑎0(𝑡) - 𝑥(2 )𝑎1(𝑡), ∈ [0, 2 ], ∈ 𝐷(𝐴). As a method of studying the spectral properties of the operator - the similar operators method serves. One of the main results is the following theorem. Theorem 3. Let functions 𝑎0 and 𝑎1 of bounded variation on a segment [0, 2 ] and sequences 1, 2 : N → R+ = [0,∞) defined by formulas: 1(𝑛) = (︃ 20 𝑛4 + 1 𝑛6 + 4 𝑛2 Σ︁ 𝑚≥1 𝑚̸=𝑛 𝑛4 + 𝑚4 𝑚2|𝑛2 - 𝑚2|2 )︃1/2