Sturm–Liouville Problems with Transfer Condition Herglotz Dependent on the Eigenparameter: Hilbert Space Formulation

Abstract

We consider a Sturm-Liouville equation ℓy := −y ′′ + qy = λy on the intervals (−a, 0) and (0, b) with a, b > 0 and q ∈ L 2 (−a, b). We impose boundary conditions y(−a) cos α = y ′ (−a) sin α, y(b) cos β = y ′ (b) sin β, where α ∈ [0, π) and β ∈ (0, π], together with transmission conditions rationally-dependent on the eigenparameter viawith bi, aj > 0 for i = 1, . . . , N, and j = 1, . . . , M . Here we take η, κ ≥ 0 and N, M ∈ N0. The geometric multiplicity of the eigenvalues is considered and the cases in which the multiplicity can be 2 are characterized. An example is given to illustrate the cases. A Hilbert space formulation of the above eigenvalue problem as a self-adjoint operator eigenvalue problem in L 2 (−a, b) C N * C M * , for suitable N * , M * , is given. The Green's function and the resolvent of the related Hilbert space operator are expressed explicitly.Mathematics Subject Classification (2010). Primary 34B24; Secondary 34A36, 34B07.

Inverse Spectral Problem for Sturm-Liouville Operator with Boundary and Jump Conditions Dependent on the Spectral Parameter

“…Through inducting the generalized normal constants they have proved the uniqueness theorem, then a construction method for solving this inverse problem was given [30]. In 2018 and 2021, Bartels et al presented Sturm-Liouville problems with transfer condition Herglotz dependent on the eigenparameter, and showed the Hilbert space formulation of the problem and calculated out the eigenvalue and eigenfunction asymptotic formula on this problem [31] [34]. Zhang et al studied the finite spectrum of Sturm-Liouville problems with both jump conditions dependent on the spectral parameter [35].…”

Inverse problems for discontinuous Dirac operator with eigenparameter dependent boundary and transmission conditions

“…Moreover, direct and inverse spectral problems for the appearance of the parameter in boundary and transmission conditions as non-linearly have been investigated fairly completely for classical Sturm-Liouville operator and also Dirac operator [16][17][18][19][20][21][22][23][24]. Besides all this, in recent years, Sturm-Liouville operators with eigenparameter dependent discontinuity conditions were studied in [25]. And also, complete solutions of various direct and inverse spectral problems for Sturm-Liouville operators with boundary conditions containing rational Herglotz-Nevanlinna functions of the eigenvalue parameter were provided in [26] and [27], and were subsequently extended to distributional [28] and Bessel-type potentials [29,33].…”

On extensions of symmetric operators

“…There is a widely used linearization technique in the theory of Sturm-Liouville problems with boundary and/or discontinuity conditions polynomially dependent on the eigenvalue parameter. One considers a Hilbert (or Pontryagin) space of the form L 2 ⊕ C k and constructs a self-adjoint operator in this space such that the eigenvalue problem for this operator and the original boundary value problem become equivalent, in the sense that their eigenvalues coincide, the eigenfunctions of the latter problem are in one-to-one correspondence with the first components of the eigenvectors of the former problem, and so on (see, e.g., [1], [2], [8], and the references therein). Fulton [6, Remark 2.1] attributes this technique to Friedman [5, pp.…”

“…Let A be a densely defined, closed symmetric operator in a separable Hilbert space H with deficiency indices (1,1). Let {C, Γ 0 , Γ 1 } be a boundary triplet for A * .…”