We investigate the spectral properties of Sturm-Liouville operators with measure potentials. We obtain two-sided estimates for the spectral distribution function of the eigenvalues. As a corollary, we derive a criterion for the discreteness of the spectrum and a criterion for the membership of the resolvents to Schatten classes. We give two side estimates for the lower bound of the essential spectrum. Our main tool in achieving this is Otelbaev's function.
We study asymptotics of eigenvalues, eigenfunctions and norming constants of singular energy-dependent Sturm--Liouville equations with complex-valued potentials. The analysis essentially exploits the integral representation of solutions, which we derive using the connection of the problem under study and a Dirac system of a special form.
We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.
The Sturm--Liouville problem $-y''-λρy=0$, $y(0)=y(1)=0$, where $ρ$ is a generalized derivative of self-similar function $P\in L_2[0,1]$ with spectral degree D=0, is studied. Asymptotic formulas for eigenvalues are obtained.
We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth order Sturm-Liouville problem near a singular endpoint. Of particular interest are the results in the so called lim-3 case, which has no analogue in second order singular problems.
We consider the Sturm-Liouville operator Lu=u''-q(x)u with periodic or antiperiodic boundary conditions. It is shown that depending of Fourier coefficients of the potential q(x) the system of root functions may have or may not have the basis property.
Selfadjoint Sturm-Liouville operators $H_ω$ on $L_2(a,b)$ with random potentials are considered and it is proven, using positivity conditions, that for almost every $ω$ the operator $H_ω$ does not share eigenvalues with a broad family of random operators and in particular with operators generated in the same way as $H_ω$ but in $L_2(\tilde a,\tilde b)$ where $(\tilde a,\tilde b)\subset(a,b)$.
A method of calculation of eigenvalues of the problem $-y''-λρy=0$, $y(0)=y(1)=0$, where $ρ$ is a distribution having self-similar primitive $P\in L_2[0,1]$, is described.
We give a complete description of the set of spectral data (eigenvalues and specially introduced norming constants) for Sturm--Liouville operators on the interval $[0,1]$ with matrix-valued potentials in the Sobolev space $W_2^{-1}$ and suggest an algorithm reconstructing the potential from the spectral data that is based on Krein's accelerant method.
We consider the spectral problems for the Sturm-Liouville operator generated by the Dirichlet, Neumann, Dirichlet-Neumann and Neumann-Dirichlet conditions. The necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann-Dirichlet problems is proved. Also the necessary and sufficient condition for the coincidence of the spectrum, except zero, of the Dirichlet and Neumann problems is proved. An application to periodic and anti-periodic problems is given.
The paper investigates spectral properties of multi-interval Sturm-Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the trace class and for the systems of root functions to be complete are found. Results of paper are new for one-interval boundary value problems as well.
We study the inverse spectral problem of reconstructing energy-dependent Sturm-Liouville equations from their Dirichlet spectra and sequences of the norming constants. For the class of problems under consideration, we give a complete description of the corresponding spectral data, suggest a reconstruction algorithm, and establish uniqueness of reconstruction. The approach is based on connection between spectral problems for energy-dependent Sturm-Liouville equations and for Dirac operators of special form.
In the paper the Sturm-Liouville problem $-y''-ρy=0$, $y(0)=y(1)=0$ is studied. $ρ$ is a generalized derivative of function $P\in L_2[0,1]$. For self-similar $P$ asymptotic formulas for eigenvalues are obtained.
We study the Sturm-Liouville problem $-y''-ρy=0$, $y(0)=y(1)=0$. $ρ$ is a generalized derivative of function $P\in L_2[0,1]$. For self-similar $P$ asymptotic formulas for eigenvalues are obtained. In this paper we consider two cases of self-similarity of $P$: 1) arithmetic degenerate, and 2) non-arithmetic.
The paper deals with singular Sturm-Liouville expressions with matrix-valued distributional coefficients. Due to a suitable regularization, the corresponding operators are correctly defined as quasi-differentials. Their resolvent convergence is investigated and all self-adjoint, maximal dissipative, and maximal accumulative extensions are described in terms of homogeneous boundary conditions of the canonical form.
We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions affinely dependent on the eigenparameter. We give sufficient conditions under which the root vectors of this Sturm-Liouville problem can be selected to form a Riesz basis of a corresponding weighted Hilbert space.
In this paper we consider the Sturm-Liuoville operator in the Hilbert space $L_2$ with the singular complex potential of $W^{-1}_2$ and two-point boundary conditions. For this operator we give sufficient conditions for norm resolvent approximation by the operators of the same class.
We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.
We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.
This work deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs.
In this work, we study discontinuous Sturm-Liouville type problems with eigenparameter dependent boundary condition and transmission conditions at three interior points. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We show that the eigenvalues of the problem are analytically simple, and the eigenfunctions of A are complete in H.
We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $ω$-parametric analytic family better than order of $(ω\lnω)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $ω^{-m}$.
We investigate the discrete Sturm-Liouville equations. In the present situation, the right hand side of the equation does not give rise to a positive definite quadratic form and we use instead the left hand side to define such a form. We prove in this paper that this form determines a Hilbert space (such problems are called left-definite).
