An inverse problem for non-selfadjoint Sturm–Liouville operator with discontinuity conditions inside a finite interval

Inverse spectral problems for non-self-adjoint Sturm–Liouville operators with discontinuous boundary conditions

“…(3) The symbol m n denotes the algebraic multiplicity of the eigenvalue λ n , n ∈ S B , and m ∞ n denotes the algebraic multiplicity of λ ∞ n , n ∈ S B ∞ . For sufficiently large n it is well known that m ∞ n = m n = 1 (see Lemma 2.3 in [17]). Now we turn to give the definition of the generalized norming constants for the problem B. Denote (1.7) κ n+ν := ϕ n+ν (π) , α n+ν :=…”

“…Recall that σ (B) := {λ n } n∈N0 and σ (B ∞ ) := {λ ∞ n } n∈N0 are the sequences consisting of all the eigenvalues of B and B ∞ , respectively. By the asymptotics of the eigenvalues λ n and λ ∞ n [17], it is easy to see that there exist constants r 1 and r 2 such that min…”

“…To distinguish κ n and α n , in this paper, κ n is called the generalized ratio, and α n is called the generalized normalizing constant. Moreover, it follows from [17,Theorem 4.1] that for n ∈ S B , ν = 0, 1, . .…”

“…Assume that ϕ (x, λ), ψ (x, λ) , ψ ∞ (x, λ) are solutions of the equation (3) The symbol m n denotes the algebraic multiplicity of the eigenvalue λ n , n ∈ S B , and m ∞ n denotes the algebraic multiplicity of λ ∞ n , n ∈ S B ∞ . For sufficiently large n it is well known that m ∞ n = m n = 1 (see Lemma 2.3 in [17]). Now we turn to give the definition of the generalized norming constants for the problem B. Denote Then κ n and α n , n ∈ N 0 , are called the generalized norming constants corresponding to λ n .…”

“…For example, discontinuous inverse problems appear in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics [1,2]. In the last decades, inverse spectral problems for Sturm-Liouville operators with different type discontinuities have attracted tremendous interest [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. These start with the fundamental work given by V. Ambarzumian [19] and then by G. Borg [20], B. Levitan [21,22], and V. Marchenko [23,24] for the classical Sturm-Liouville operators.…”