Related papers: Semigroups for One-Dimensional Schr\"odinger Opera…

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB - BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace…

Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild regularity…

Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-\Delta +V(x)$ be a Schr\"odinger operator on $\Omega$ with potential V locally bounded. We seek the weakest conditions we can find on the…

We study Schr\"odinger operators $H:= -\Delta + V$ with potentials $V$ that have power-law growth (not necessarily polynomial) at 0 and at $\infty$ using methods of Lie theory (Lie-Rinehart algebras) and microlocal analysis. More precisely,…

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}…

Semilinear hyperbolic stochastic partial differential equations (SPDEs) find widespread applications in the natural and engineering sciences. However, the traditional Gaussian setting may prove too restrictive, as phenomena in mathematical…

We study Schr\"odinger operators $H=-\Delta+V$ in $L^2(\Omega)$ where $\Omega$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential…

The present paper is devoted to the study of resonances for a $1$D Schr\"{o}dinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb N}_{L}= -\Delta +…

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator $T$ on a separable infinite-dimensional Hilbert space $H$ such that the orbit $\{T^{n}x;\ n\ge 0\}$ of every non-zero vector $x\in…

We consider semiclassical Schr\"odinger operators acting in $L^2(\mathbb{R}^d)$ with $d\geq3$. For these operators we establish a sharp spectral asymptotics without full regularity. For the counting function we assume the potential is…

The present paper addresses questions on resonances for a $1$D Schr\"{o}dinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb N}_L = -\Delta + V…

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line,…

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2\Delta+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is…

A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators id\pi(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the…

We consider a semi-classical Schr\"odinger operator, -h^2\Delta + V(x). Assuming that the potential admits a unique global minimum and that the eigenvalues of the Hessian are linearly independent over the rationals, we show that the…

We consider fractional Schr\"odinger operators $H=(-\Delta)^\alpha+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2\alpha$, $\alpha>1$. We show that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for…

This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a…

The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave…

We study the time-dependent Schr\"odinger operator $P = D_t + \Delta_g + V$ acting on functions defined on $\mathbb{R}^{n+1}$, where, using coordinates $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $D_t$ denotes $-i \partial_t$, $\Delta_g$…

For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space $\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator for the Schr\"odinger equation $i\psi'_t=\mathcal{H}\psi$, i.e. if $\psi(0,x)=\psi_0(x)$…