Related papers: Uniform asyptotic formulae for eigenfunctions of S…

In this paper we study a Sturm--Liouville operator $Ly=-y"+q(x)y$ in the space $L_2[0,\pi]$ with Direchlet boundary conditions. Here the potential $q$ is a first order distribution: $q\in W_2^{-1}[0,\pi]$. Such operators were defined in our…

We deal with the Sturm--Liouville operator $Ly=l(y)=-\dfrac{d^2y}{dx^2}+q(x)y,$ with Dirichlet--Neumann boundary conditions $ y(0)=y'(\pi)=0 $ in the space $L_2[0,\pi]$. We assume that the potential $q$ is complex-valued and has the form…

We study asymptotic behavior of the eigenvalues of Strum--Liouville operators $Ly= -y'' +q(x)y $ with potentials from Sobolev spaces $W_2^{\theta -1}, \theta \geqslant 0$, including the non-classical case $\theta \in [0,1)$ when the…

We consider a Sturm--Liouville operator $Ly=-y''+qy$ in $L_2[0,\pi]$ with Dirichlet boundary conditions. We assume, that the potential $q$ is complex valued and belongs to Sobolev space $W_2^\theta[0,\pi]$, $\theta\in(-1,-1/2$. This…

We obtain the uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L_{t}(q) with a potential q\inL_{1}[0,1] and with t-periodic boundary conditions, t\in(-{\pi},{\pi}]. Using these formulas, we…

We derive eigenvalue asymptotics for Sturm--Liouville operators with singular complex-valued potentials from the space $W^{\al-1}_{2}(0,1)$, $\al\in[0,1]$, and Dirichlet or Neumann--Dirichlet boundary conditions. We also give application of…

It is well known that a potential $q$ of the Sturm-Liouville operator $Ly= -y" +q(x)y$ on the finite interval $[0, \pi]$ can be uniquely recovered by the spectrum $\{\lambda_k\}_1^\infty$ and norming constants $\{\alpha_k\}_1^\infty$ of…

Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$. Let $\{\lambda_k\}_1^\infty$ and $\{\alpha_k\}_1^\infty$ be the sequences of the eigenvalues…

The paper deals with two inverse problems for Sturm--Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map $F:\,…

We consider a Sturm--Liouville $Ly=-y''+q(x)y$ in space $L_2[0,\pi]$ with potential from Sobolev space $W_2^{-1}[0,\pi]$. Moreover, we assume, that $q=u'$, where $u\in L_2[0,\pi]$. We consider Direchlet boundary conditions $y(0)=y(\pi)=0$,…

In this paper we propose four different methods to determine Sturm-Liouville operator on an interval $(a,b)$ in case, when a potential $q(x)$ is a distribution from the Sobolev space with negative index of smoothness, i.e. (q\in…

We introduce a novel approach for dealing with eigenvalue problems of Sturm-Liouville operators generated by the differential expression \begin{equation*} Ly=\frac{1}{r}\left( -(p\left[ y^{\prime }+sy\right] )^{\prime }+sp\left[ y^{\prime…

We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root…

We obtain sharp asymptotic formulas for the eigenvalues and norming constants of Sturm-Liouville operators associated with the differential expression \[ -\frac{d^2}{dx^2} + x + q(x), \quad x\in [0,\infty), \] together with the boundary…

We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with general regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root…

We obtain asymptotic representations as $\lambda \to \infty$ in the upper and lower half-planes for the solutions of the Sturm--Liouville equation $$ -y"+p(x)y'+q(x)y= \lambda ^2 \rho(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ under the…

Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem $-y" + q \left( x \right) y = \mu y,$ $y \left(0\right)=0,$ $y\left( \pi \right)\cos \beta + y'\left( \pi…

The indefinite Sturm-Liouville operator $A = (\sgn x)(-d^2/dx^2+q(x))$ is studied. It is proved that similarity of $A$ to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of…

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in space of vector-functions by the Sturm-Liouville equation with m by m matrix potential and the boundary…

In this paper, by using the similar methods of [O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694] we extend some…