Related papers: Eigenvalue bounds for Schr\"odinger operators with…

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger…

We prove that the low energy parts of the wave operators $W_\pm$ for Schr\"odinger operators $H = -\lap + V(x)$ on $\R^4$ are bounded in $ L^p(\R^4)$ for $1<p\leq 2$ and are unbounded for $2<p\leq \infty$ if $H$ has resonances at the…

We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $\Omega\subset \mathbb{R}^N…

We introduce a notion of maximal potentials and we prove that they form bounded operators from $L^p$ to the homogeneous Sobolev space $\dot{W}^{1,p}$ for all $n/(n-1)<p<n$. We apply this result to the problem of boundedness of the spherical…

We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schr\"odinger operators $-\Delta + V$ with nonnegative potential $V$. These functions are proved to be bounded on $L^p$ for all $p \in (1, 2]$. The…

We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for…

Non-self-adjoint Schrodinger operators which correspond to non-symmetric zero-range potentials are investigated. We show that various properties of these operators (eigenvalues, exceptional points, spectral singularities and the property of…

We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schr\"odinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the…

Using purely physical arguments it is claimed that for ID Schrodinger operators with complex PT- Symmatric potentials having a purely real attractive potential well and a purely imaginary repulsive part,bound state eigenvalues will be…

We consider vector-valued magnetic Schr\"odinger operators $-\bm \Delta_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries…

This article deals with the inverse problem of determining the unbounded real-valued electric potential of the Robin Laplacian on a bounded domain of dimension 3 or greater, by incomplete knowledge of its boundary spectral data. Namely, the…

We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the…

We relax the regularity condition on potentials of the Schr\"odinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [11] and [5].

For general non-symmetric operators $A$, we prove that the moment of order $\gamma \ge 1$ of negative real-parts of its eigenvalues is bounded by the moment of order $\gamma$ of negative eigenvalues of its symmetric part $H = {1/2} [A +…

In this work we investigate a class of degenerate Schr\"odinger equations associated to degenerate elliptic operators with irregular potentials on $\Ran$ by introducing a suitable H\"ormander metric $g$ and a $g$-weight $m$. We establish…

We show various $L^p$ estimates for Schr\"odinger operators $-\Delta+V$ on $\RR^n$ and their square roots. We assume reverse H\"older estimates on the potential, and improve some results of Shen \cite{Sh1}. Our main tools are improved…

We show that phase space bounds on the eigenvalues of Schr\"{o}dinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In…

We consider Schr\"odinger operators on [0,\infty) with compactly supported, possibly complex-valued potentials in L^1([0,\infty)). It is known (at least in the case of a real-valued potential) that the location of eigenvalues and resonances…

We prove that the wave operators for $n \times n$ matrix Schr\"odinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces $L^p(\mathbb R^+, \mathbb C^n), 1 < p < \infty, $ for slowly decaying…

This paper establishes the $L^p$ boundedness of wave operators for linear Schr\"odinger equations in $\mathbb{R}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application…