Related papers: Hadamard-type formulas via the Maslov form

This paper is dedicated to $L^p$ bounds on eigenfunctions of a Sch\"odinger-type operator $(-\Delta_g)^{\alpha/2} +V$ on closed Riemannian manifolds for critically singular potentials $V$. The operator $(-\Delta_g)^{\alpha/2}$ is defined…

In this paper, following the works on non-harmonic analysis of boundary value problems by Tokmagambetov, Ruzhansky and Delgado, we use Operator Ideals Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of…

We derive conditions that ensure the existence of a bounded $H_\infty$-calculus in weighted $L_p$-Sobolev spaces for closed extensions $\underline{A}_T$ of a differential operator $A$ on a conic manifold with boundary, subject to…

The Hadamard well-posedness of the nonlinear Schr\"odinger equation with power nonlinearity formulated on the spatial quarter-plane is established in a low-regularity setting with Sobolev initial data and Dirichlet boundary data in…

Let $\Omega\subset\RR^n$ ($n\ge 1$) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms, we give a rigorous characterization of the realization in $L^2(\Omega)$ of…

We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain $D$ of ${\mathbb R}^n$ for a second order parameter-dependent elliptic differential operator $A (x,\partial, \lambda)$ with complex-valued essentially…

In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic…

A flag domain in $\mathbb{R}^{3}$ is a subset of $\mathbb{R}^{3}$ of the form $\{(x,y,t) : y < A(x)\}$, where $A \colon \mathbb{R} \to \mathbb{R}$ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic…

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the…

Consider operators $L_{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^N$. Assume that $V\in C^\alpha(\Omega)$ satisfies $|V(x)| \leq \bar a\,\mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and that $L_V$ has a…

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the…

In a bounded domain $G$ with smooth border studied boundary value and spectral problems for operators of the rotor (vortex) and the gradient of the divergence $+\lambda\,I$ in the Sobolev spaces. For $\lambda\neq 0$ these operators are…

We consider, for $h, E > 0$, resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V - E$. Near infinity, the potential takes the form $V = V_L+ V_S$, where $V_L$ is a long range potential which is Lipschitz with…

In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$(P_{\Omega}) \left\{…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $\mu\leq (\frac{N-k-2}{2})^2$, put…

Using an extension of the H\"ormander product of distributions, we obtain an intrinsic formulation of one-dimensional Schr\"odinger operators with singular potentials. This formulation is entirely defined in terms of standard {\it Schwartz}…

This work is about global H\"older regularity for solutions to elliptic partial differential equations subject to mixed boundary conditions on irregular domains. There are two main results. In the first, we show that if the domain of the…

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that…

The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$…