Related papers: Regularization of singular Sturm-Liouville equatio…

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…

Sturm-Liouville spectral problem for equation $-(y'/r)'+qy=\lambda py$ with generalized functions $r\ge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $r\equiv 1$. The case of $q=0$ and…

We propose a regularization of the formal differential expression of order $m \geqslant 3$ $$ l(y) = i^my^{(m)}(t) + q(t)y(t), \,t \in (a, b), $$ applying quasi-derivatives. The distribution coefficient $q$ is supposed to have an…

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 introduce and investigate symmetric operators $L_0$ associated in the complex Hilbert space $L^2(\mathbb{R})$ with a formal differential expression \[l[u] :=-(pu')'+qu + i((ru)'+ru') \] under minimal conditions on the regularity of the…

We give explicit formulas for a pair of linearly independent solutions of $(py')'(x)+q(x)=(\lambda_1r_1(x)+\cdots+\lambda_dr_d(x))y(x)$, thus generalizing to arbitrary $d$ previously known formulas for $d=1$. These are power series in the…

We consider the boundary problem -y''(x)+q(x)y(x)=f(x), lim_{|x|\to\infty}y^{(i)}(x)=0, i=0,1, where f(x)\in L_p(R), p\in[1,\infty], 1\le q(x)\in L_1^{\loc}(R). For this boundary problem we obtain: 1) necessary and sufficient conditions for…

In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized functions $q,r\in W_2^{-1}[0,1]$…

In this paper we consider the following Sturm-Liouville equation \[ \left\{ \begin{aligned} -(x^{2\alpha}u'(x))'+u(x)&=f(x) && \text{in } (0,1],\\ u(1)&=0 \end{aligned} \right. \] where $\alpha<1$ is a nonzero real number and $f$ belongs to…

Any self-adjoint extension of a (singular) Sturm-Liouville operator bounded from below uniquely leads to an associated sesquilinear form. This form is characterized in terms of principal and nonprincipal solutions of the Sturm-Liouville…

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…

We consider a regular singular Sturm-Liouville operator $L:=-\frac{d^2}{dx^2} + \frac{q(x)}{x^2 (1-x)^2}$ on the line segment $[0,1]$. We impose certain boundary conditions such that we obtain a semi-bounded self-adjoint operator. It is…

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 fitst order distribution $q\in W_2^{-1}[0,\pi]$. Such operators were defined in our…

We consider difference operators in $L^2$ on $\R$ of the form $$ L f(s)=p(s)f(s+i)+q(s) f(s)+r(s) f(s-i) ,$$ where $i$ is the imaginary unit. The domain of definiteness are functions holomorphic in a strip with some conditions of decreasing…

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 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,…

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…

In this paper, Sturm-Liouville problem for difference equations is considered with potential function q(n). The representations of solutions are obtained by variation of parameters method. These solutions are proved, using summation by…

In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized function $q\in W_2^{-1}[0,1]$ is…

Consider the operator $H\p=-\p''+q\p=\l\p$, $\p(0)=0$, $\p'(1)+b\p(1)=0$ acting in $L^2(0,1)$, where $q\in L^2(0,1)$ is a real potential. Let $\l_n(q,b)$, $n\ge 0$, be the eigenvalues of $H$ and $\n_n(q,b)$ be the so-called norming…