Related papers: Uniform asyptotic formulae for eigenfunctions of S…

Inverse spectral problems for Sturm-Liouville operators with nonlocal boundary conditions are studied. As the main spectral characteristics we introduce the so-called Weyl-type function and two spectra, which are generalizations of the…

In this paper, we obtain a necessary and sufficient condition for $L^{\infty}$-uniqueness of Sturm-Liouville operator $a(x)\frac{d^2}{dx^2} + b(x) \frac d{dx} -V$ on an open interval of $\rr$, which is equivalent to the $L^1$-uniqueness of…

The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behaviour of the first eigenpair (ground state) of this…

In this paper we consider conformal spectral estimates of the Dirichlet-Laplace operator in conformal regular domains $\Omega \subset \mathbb R^2$. This study is based on the geometric theory of composition operators on Sobolev spaces that…

Consider the minimal Sturm-Liouville operator $A = A_{\rm min}$ generated by the differential expression $\mathcal{A} := -\frac{d^2}{dt^2} + T$ in the Hilbert space $L^2(\mathbb{R}_+,\mathcal{H})$ where $T = T^*\ge 0$ in $\mathcal{H}$. We…

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…

In this article we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator $\sum_{i=1}^{d}(-\partial_i^2)^{s}, \, s\in (1/2,1]$ on an open and bounded subdomain $\Omega \subset \mathbb{R}^d$ and predict…

Inverse nodal problem on diffusion operator is the problem of finding the potential functions and parameters in the boundary conditions by using nodal data. In particular, we solve the reconstruction and stability problems using nodal set…

We study self-adjoint extensions of a second order differential operator of Sturm-Liouville type on a graph. We relate self-adjointness of the operator to the existence of non-complete trajectories of the Hamiltonian vector field defined by…

For the classical Sturm-Liouville operators, we prove the sharp bounds for all nodes of eigenfunctions by regarding these nodes as nonlinear functionals of potential $q\in L^1[0,1]$. By studying the optimization problems to minimize or to…

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(\lambda)= \kappa_0\lambda^d…

Let $\dot A$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by $\mathcal{N}_i = \ker \big(\big(\dot A\big)^* - i I_{\mathcal{H}}\big)$, $\dim…

We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $\Omega \subset \mathbb C$. The…

We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to…

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We…

The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and…

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 this work we consider the following $\alpha$-stable-like operator (a class of pseudo-differential operator) $$ {\mathscr L} f(x):=\int_{\mathbb R^d}[f(x+\sigma_x y)-f(x)-1_{\alpha\in[1,2)}1_{|y|\leq 1}\sigma_x y\cdot\nabla f(x)]\nu_x(d…

In this paper, we explore the inverse spectral problem of Sturm-Liouville operator on a star-like graph. To this fixed star-like graph centered at the origin as its vertex, we attach $m$ edges. On each edge, we impose the Sturm-Liouville…

We study one-dimensional Schr\"{o}dinger operators $\mathrm{S}(q)$ on the space $L^{2}(\mathbb{R})$ with potentials $q$ being complex-valued generalized functions from the negative space $H_{unif}^{-1}(\mathbb{R})$. Particularly the class…