Related papers: Improved sharp spectral inequalities for Schr\"odi…

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for fourth order Schr\"odinger operators in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of…

Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems,…

We establish various $L^{p}$ estimates for the Schr\"odinger operator $-\Delta+V$ on Riemannian manifolds satisfying the doubling property and a Poincar\'e inequality, where $\Delta $ is the Laplace-Beltrami operator and $V$ belongs to a…

Let $(X,d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $\mu $ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite…

In this article, we introduce a new methodology to prove global parabolic Harnack inequalities on Riemannian manifolds. We focus on presenting a new proof of the global pointwise Harnack inequality satisfied by positive solutions of the…

We consider operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$ that locally behave as a magnetic Schr\"odinger operator. For the magnetic Schr\"odinger operators we suppose the magnetic potentials are smooth and the electric potential is…

We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than…

We establish $\frac{1}{2}$-H\"older continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schr\"odinger operators under suitable boundary conditions and exponentially decaying small potentials. These…

We prove a Lieb-Thirring type inequality for a complex perturbation of a d-dimensional massive Dirac operator $D_m, m\geq 0$ whose spectrum is $]-\infty , -m]\cup[m , +\infty[$. The difficulty of the study is that the unperturbed operator…

We consider the Schr\"odinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we…

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an…

We consider Sch\"odinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta +…

We study the 1-D Schr\"odinger operators in Hilbert space $L^{2}(\mathbb{R})$ with real-valued Radon measure $q'(x)$, $q\in \mathrm{BV}_{loc}(\mathbb{R})$ as potentials. New sufficient conditions for minimal operators to be bounded below…

We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms…

We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we…

We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition in non-Lipschitz domains $\widetilde{\Omega} \subset \mathbb C$. The suggested method is…

We prove a general Levinson's theorem for Schr\"odinger operators in two dimensions with threshold obstructions at zero energy. Our results confirm and simplify earlier seminal results of Boll\'e, Gesztesy et al., while providing an…

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schroedinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Soerensen,…

With derive sharp spectral asymptotics (with the remainder estimate $O(\mu ^{-1}h^{1-d}+\mu ^{\frac{d} {2}-1}h^{1-\frac{d}{2}})$ for $d$-dimensional Schr\"odinger operator with a strong magnetic field; here $h$ and $\mu$ are Plank and…

In this article we study the semiclassical spectral measures associated with Schr\"odinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give…