Related papers: A complete classification of threshold properties …

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a…

The homogenization of eigenvalues of non-Hermitian Maxwell operators is studied by the H-convergence method. It is assumed that the Maxwell systems are equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or…

Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $ -\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and…

In this paper, we investigate the delocalization property of the discrete Schr\"odinger operator $H_\omega=-\Delta+v_n\omega_n\delta_{n,n'}$, where $v_n=\kappa |n|^{-\alpha}$ and $\omega=\{\omega_n\}_{n\in\mathbb{Z}^d}\in \{\pm…

In the present paper we investigate the set $\Sigma_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space…

We investigate the spectral properties of self-adjoint Schr\"odinger operators with attractive $\delta$-interactions of constant strength $\alpha > 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum…

We prove Strichartz estimates for the Schroedinger operator $H = -\Delta + V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from…

We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices $(1, 1)$.

We consider the Schr{\"o}dinger operator $-\Delta +V(x)$ in $L^2({\bf R}^3)$ with a real short-range (integrable) potential $V$. Using the associated Fredholm determinant, we present new trace formulas, in particular, the ones in terms of…

In the paper we consider a functional-difference operator $H=U+U^{-1}+V$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{\pi i\tau}$ and $\tau>0$. The operator $H$ has applications in the conformal…

Let $S\subset\mathbb{R}^3$ be a $C^4$-smooth relatively compact orientable surface with a sufficiently regular boundary. For $\beta\in\mathbb{R}_+$, let $E_j(\beta)$ denote the $j$th negative eigenvalue of the operator associated with the…

We prove Kenig--Ruiz--Sogge type uniform resolvent estimates for selfadjoint magnetic Schr\"{o}dinger operators $H=(i\partial+A(x))^2+V(x)$ on $\mathbb{R}^{n}$, $n\ge3$. Under suitable decay assumptions on the electric and magnetic…

Let Lf(x)=-\Delta f(x) + V(x)f(x), V\geq 0, V\in L^1_{loc}(R^d), be a non-negative self-adjoint Schr\"odinger operator on R^d. We say that an L^1-function f belongs to the Hardy space H^1_L if the maximal function M_L f(x)=\sup_{t>0}…

Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $\Phi$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated…

Let L be a Schr\"odinger operator of the form L=-\Delta+V, where the nonnegative potential V satisfies a reverse H\"older inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted H\"older…

In this paper we consider the two-dimensional Schr\"odinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the…

Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger…

The goal of this paper is the spectral analysis of the Schr\"{o}dinger operator $H=L+V$ , the perturbation of the Taibleson-Vladimirov multiplier $L=\mathcal{D}^{\alpha}$ by a potential $V$. Assuming that $V$ belonges to a class of fast…

We prove quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + Vu = \lambda u$, where $\lambda \in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle…

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…