The characteristic of the spectrum of some boundary-value problems for the Sturm-Liouville operator

Inverse problems of spectral analysis for the Sturm-Liouville operator with regular boundary conditions: I

“…where f (µ) ∈ P W − π . Taking into account (5), using that the numbers cn are the roots of equation ( 45), repeating arguments of Theorem 2, we obtain ∆(µ n ) = ∆(µ n ). Then, as for Theorem 2, we get ∆(µ) ≡ ∆(µ).…”

“…If a 14 is a real number and q(x) is a real-valued function, then the corresponding boundary value problem is selfadjoint [2]. For this case inverse problem was studied in [5], [6]. Exhausting description of boundary conditions of type III and IV is given in [2].…”

“…Obviously, the spectrum of problem (1)+( 2) is defined uniquely by a quadruple of parameters (α, γ, θ, q(x)). The characteristic determinant ∆(µ) of problem (1)+(2) defined by formula (5) is called the characteristic determinant corresponding to a collection (α, γ, θ, q(x)). Throughout the following ||f || stands for ||f || L 2 (0,π) , < q >= 1 π π 0 q(x)dx.…”

“…where f (µ) ∈ P W − π . Taking into account (5) and remembering that the numbers c n are the roots of equation ( 17), we obtain…”

“…This completes the proof of lemma 3. It follows from equality (5), asymptotic formula (28) and the Hadamard theorem [16, p. 259] that…”