Inverse Problems of Spectral Analysis for Sturm-Liouville Operators With Nonseparated Boundary Conditions

The reconstruction of a differential operator by its spectrum

“…Therefore, the sequence {A,} satisfies the conditions of Theorem 1, whence {An} is the spectrum of the boundary-value problem generated by the operator l with a real potential q(x) 9 L~[0, ~r] and by boundary conditions (6). It is easy to show that the characteristic function of the problem of the form (1), (2) with the reconstructeded potential coincides with Al(z).…”

“…Remark. The proof of the sufficiency of the hypotheses of Theorem 1 (which gives the solution of the inverse problem by a spectrum) can be reduced to the problem of reconstruction of the operator l by the spectrum and normalization constants of boundary-value problem (1), (6), solved in [4]. In fact, if we denote = (-1)n+lc~ (V~n),…”

“…Therefore, the function A(A) can be reconstructed in the form of an infinite product by its zeros -t-x/z~, =kx/-fi~-, .... Moreover, given the sequence {#,}, one can determine the constant h, since the asymptotic equality (8) holds [6] with A = ~ fo q(x) dx; then we obtain oolim [(,/.~k_l -2k + 1) (2k-1)-2k( .V~-;-2k)]. …”

“…As follows from [6], the eigenvalues ~ of problem (1), (20) Unlike boundary-values problems for Eq. (1) under separated boundary conditions, the eigenvalues of boundary-value problems of the form (1), (2) can be multiple (of multiplicity not higher than 2).…”

“…Using arguments similar to [7, Ch. 1, Theorem 4.1], one can show that an eigenvalue #k (fire) of problem (1), (2) ( (1), (20) Therefore, multiple eigenvalues of these problems are eigenvalues of boundary-value problem (1), (6). By the standard method (see, for example, [6]), one can prove that the arrangement of elements of the sequences {An}, {#n}, {~} is defined by the inequalities -cC<Ao<~o<ffo<,\l<~l<#l<A2<#2<....…”