arXiv subjects

Aleksey Kostenko

Publications and source records attributed to Aleksey Kostenko.

49 records · Page 3Linked to original sources

arXiv2013

The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions

We consider a regular indefinite Sturm-Liouville eigenvalue problem \{$-f" + q f = λr f$} on $[a,b]$ subject to general self-adjoint boundary conditions and with a weight function $r$ which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions $r$ for which the Riesz basis property can be completely characterized in terms of the local behavior of $r$ in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of $r$ in a neighborhood of the turning points, local conditions on $r$ near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.

math.SP↗

arXiv2013

1-D Schrödinger operators with local point interactions: a review

We review recent developments in the theory of 1-D Schrödinger operators with local point interactions on a discrete set. The progress in this area was stimulated by recent advances in the extension theory of symmetric operators and in the theory of ordinary differential operators with distributional coefficients.

math-ph↗

arXiv2012

Spectral theory of semibounded Schrödinger operators with $δ'$-interactions

We study spectral properties of Hamiltonians $\rH_{X,\gB,q}$ with $δ'$-point interactions on a discrete set $X={x_k}_{k=1}^\infty\subset\R_+$. %at the centers $x_n$ on the positive half line in terms of energy forms. Using the form approach, we establish analogs of some classical results on operators $\rH_q=-d^2/dx^2+q$ with locally integrable potentials $q\in L^1_{\loc}(\R_+)$. In particular, we establish analogues of the Glazman-Povzner-Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with $δ$-interactions, spectral properties of operators $\rH_{X,\gB,q}$ are closely connected with those of $\rH_{X,q}^N=\oplus_{k}\rH_{q,k}^N$, where $\rH_{q,k}^N$ is the Neumann realization of $-d^2/dx^2+q$ in $L^2(x_{k-1},x_k)$.

math-ph↗

arXiv2012

Schrödinger operators with concentric $δ$-shells

We investigate the spectral properties of the Schrödinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$ \mathbf{H}_{R,α}=-Δ+\sum_{k=1}^\inftyα_kδ(|x|-r_k). $$ We obtain necessary and sufficient conditions for the operator $\mathbf{H}_{R,α}$ to be self-adjoint, lower-semibounded. Also we investigate the spectral types of $\mathbf{H}_{R,α}$.

math-ph↗

arXiv2012

The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality

We study two problems. The first one is the similarity problem for the indefinite Sturm-Liouville operator \[ A=-(\sgn\, x)\frac{d}{wdx}\frac{d}{rdx} \] acting in $L^2_{w}(-b,b)$. It is assumed that $w,r\in L^1_{\loc}(-b,b)$ are even and positive a.e. on $(-b,b)$. The second object is the so-called HELP inequality \[(\int_{0}^b\frac{1}{\tilde{r}}|f'|\, dx)^2 \le K^2 \int_{0}^b|f|^2\tilde{w}\,dx\int_{0}^b\Big|\frac{1}{\tilde{w}}\big(\frac{1}{\tilde{r}}f'\big)'\Big|^2\tilde{w}\, dx, \] where the coefficients $\tilde{w},\tilde{r}\in L^1_{\loc}[0,b)$ are positive a.e. on $(0,b)$. Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. The main objective of the present paper is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl-Titchmarsh $m$-functions at 0 and at $\infty$. As a biproduct of this result we show that both problems are closely connected. Namely, the operator $A$ is similar to a self-adjoint one precisely if the HELP inequality with $\tilde{w}=r$ and $\tilde{r}=w$ is valid. Next we characterize the behavior of $m$-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker-Plank equation.

math.SP↗

arXiv2012

Weyl-Titchmarsh Theory for Schroedinger 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.

math.SP↗

arXiv2012

On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems

In 1996, H. Volkmer observed that the inequality \[(\int_{-1}^1\frac{1}{|r|}|f'|dx)^2 \le K^2 \int_{-1}^1|f|^2dx\int_{-1}^1\Big|\Big(\frac{1}{r}f'\Big)'\Big|^2dx \] is satisfied with some positive constant $K>0$ for a certain class of functions $f$ on $[-1,1]$ if the eigenfunctions of the problem \[ -y"=λ\, r(x)y,\quad y(-1)=y(1)=0 \] form a Riesz basis of the Hilbert space $L^2_{|r|}(-1,1)$. Here the weight $r\in L^1(-1,1)$ is assumed to satisfy $xr(x)>0$ a.e. on $[-1,1]$. We present two criteria in terms of Weyl-Titchmarsh $m$-functions for the Volkmer inequality to be valid. Using these results we show that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if $r$ is odd.

math.SP↗

arXiv2012

The similarity problem for indefinite Sturm-Liouville operators with periodic coefficients

We investigate the problem of similarity to a self-adjoint operator for $J$-positive Sturm-Liouville operators $L=\frac{1}ω(-\frac{d^2}{dx^2}+q)$ with $2π$-periodic coefficients $q$ and $ω$. It is shown that if 0 is a critical point of the operator $L$, then it is a singular critical point. This gives us a new class of $J$-positive differential operators with the singular critical point 0. Also, we extend the Beals and Parfenov regularity conditions for the critical point $\infty$ to the case of operators with periodic coefficients.

math.SP↗

arXiv2011

Commutation Methods for Schroedinger 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 Schroedinger operators (also known as Bessel operators). We also investigate the connections with the generalized Baecklund-Darboux transformation.

math.SP↗

arXiv2011

On the Singular Weyl-Titchmarsh Function of Perturbed Spherical Schroedinger Operators

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schroedinger operators (also known as Bessel operators) under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. 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.

math.SP↗

arXiv2010

Inverse Eigenvalue Problems for Perturbed Spherical Schroedinger Operators

We investigate the eigenvalues of perturbed spherical Schrödinger operators under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show that the square roots of eigenvalues are given by the square roots of the unperturbed eigenvalues up to an decaying error depending on the behavior of $q(x)$ near $x=0$. Furthermore, we provide sets of spectral data which uniquely determine $q(x)$.

math.SP↗

arXiv2009

1--D Schrödinger operators with local interactions on a discrete set

Spectral properties of 1-D Schrödinger operators $\mathrm{H}_{X,α}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}α_nδ(x-x_n)$ with local point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty$ are well studied when $d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We consider $\mathrm{H}_{X,α}$ in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of $\mathrm{H}_{X,α}$ like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators $\mathrm{H}_{X,α}$ to be self-adjoint, lower-semibounded, and discrete in the case $d_*=0$. The operators with $δ'$-type interactions are investigated too. The obtained results demonstrate that in the case $d_*=0$, as distinguished from the case $d_*>0$, the spectral properties of the operators with $δ$ and $δ'$-type interactions are substantially different.

math.SP↗

arXiv2009

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 A. Boutet de Monvel and D. Shepelsky.

nlin.SI↗