Inverse spectral problem for Jacobi operators and Miura transformation

We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.

Inverse problems for a conformable fractional Sturm–Liouville operator

In this paper, a Sturm–Liouville boundary value problem which includes conformable fractional derivatives of order α, {0<\alpha\leq 1} is considered. We give some uniqueness theorems for the solutions of inverse problems according to the Weyl function, two given spectra and classical spectral data. We also study the half-inverse problem and prove a Hochstadt–Lieberman-type theorem.

On singular spectrum of finite dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)

The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations à of an operator A0 = A0* defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions AB of a symmetric operator A acting in H in terms of the Weyl function M(·) of the pair {A, A0} and boundary n-dimensional operator B = B*. Assuming that the multiplicity of singular spectrum of A0 is maximal it is established orthogonality of singular parts EsAв and EsAo of the spectral measures EAв and EAo of the operators AB and A0, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.

On the determination of differential pencils with nonlocal conditions

Inverse problems for differential pencils with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg’s inverse problem for the classical Sturm–Liouville operators.

Some analytic properties of the Weyl function of a closed linear relation

Let $L$ and $L_{0}$, where $L$ is an expansion of $L_{0}$, be closed linear relations (multivalued operators) in a Hilbert space $H$. In terms of abstract boundary operators (i.e. in the form which in the case of differential operators leads immediately to boundary conditions) some analytic properties of the Weyl function $M(\lambda)$ corresponding to a certain boundary pair of the couple $(L, L_{0}),$ are studied. In particular, applying Hilbert resolvent identity for relations, the criterion of invertibility in the algebra of bounded linear operators in $H$ for transformation $M(\lambda) - M(\lambda_{0})$ in certain small punctured neighbourhood of $\lambda_{0} $ is established. It is proved that in this case $\lambda _{0}$ is a first-order pole for the operator-function $\left(M(\lambda )- M(\lambda_{0} )\right)^{-1} $. The corresponding residue and Laurent series expansion are found. Under some additional assumptions, the behaviour of so called $\gamma$-field $Z_{\lambda}$ (being an operator-function closely connected to $M(\lambda)$) as $\lambda \to - \infty $ is investigated.

Quasi boundary triples and semi-bounded self-adjoint extensions

In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second-order partial differential equations on domains with non-compact boundaries.