Aleksey Kostenko

LONG-TIME ASYMPTOTICS FOR THE CAMASSA-HOLM EQUATION

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Camassa–Holm equation for decaying initial data, completing previous results by Boutet de Monvel and Shepelsky.

Weyl–Titchmarsh Theory for Schrödinger Operators with Strongly Singular Potentials

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schroedinger operators.

On the singular Weyl–Titchmarsh function of perturbed spherical Schrödinger operators

We investigate the singular Weyl–Titchmarsh m-function of perturbed spherical Schrodinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x)∈L1(0,1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.

Inverse eigenvalue problems for perturbed spherical Schrödinger operators

We investigate the eigenvalues of perturbed spherical Schr?dinger operators under the assumption that the perturbation q(x) satisfies xq(x) L1(0, 1). We show that the square roots of eigenvalues are given by the square roots of the unperturbed eigenvalues up to a decaying error depending on the behavior of q(x) near x = 0. Furthermore, we provide sets of spectral data which uniquely determine q(x).

Spectral theory of semibounded Sturm-Liouville operators with local interactions on a discrete set

We study the Hamiltonians HX,α,q with δ-type point interactions at the centers xk on the positive half line in terms of energy forms. We establish analogs of some classical results on operators Hq=−d2/dx2+q with locally integrable potentials q∊Lloc1(R+). In particular, we prove that the Hamiltonian HX,α,q is self-adjoint if it is lower semibounded. This result completes the previous results of Brasche [“Perturbation of Schrodinger Hamiltonians by measures—selfadjointness and semiboundedness,” J. Math. Phys. 26, 621 (1985)] on lower semiboundedness. Also we prove the analog of Molchanov’s discreteness criteria, Birman’s result on stability of a continuous spectrum, and investigate discreteness of a negative spectrum. In the recent paper [Kostenko, A. and Malamud, M., “1–D Schrodinger operators with local point interactions on a discrete set,” J. Differ. Equations 249, 253 (2010)], it was shown that the spectral properties of HX,α≔HX,α,0 correlate with the corresponding spectral properties of a certain clas...

The Similarity Problem for J-nonnegative Sturm-Liouville Operators

Abstract Sufficient conditions for the similarity of the operator A : = 1 r ( x ) ( − d 2 d x 2 + q ( x ) ) with an indefinite weight r ( x ) = ( sgn x ) | r ( x ) | are obtained. These conditions are formulated in terms of Titchmarsh–Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm–Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm–Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r ( x ) = sgn x and q ∈ L 1 ( R , ( 1 + | x | ) d x ) , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct q ∈ ⋂ γ 1 L 1 ( R , ( 1 + | x | ) γ d x ) such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q ≡ 0 , we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward–backward” diffusion equations.

Commutation methods for Schrödinger operators with strongly singular potentials

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrodinger operators (also known as Bessel operators). We also investigate the connections with the generalized Backlund–Darboux transformation.

Spectral Asymptotics for Perturbed Spherical Schrödinger Operators and Applications to Quantum Scattering

We find the high energy asymptotics for the singular Weyl–Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators).We apply this result to establish an improved local Borg–Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.

An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa–Holm equation is integrable by the inverse spectral transform in the multi-peakon case.

Indefinite Sturm–Liouville operators with the singular critical point zero

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh