Related papers: Logarithmic Schr\"odinger operators

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 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 prove upper and lower bounds for sums of eigenvalues of Lieb-Thirring type for non-self-adjoint Schr\"odinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be…

Let $L=-\Delta + V(x)$ be a Schr\"odinger operator on $\mathbb R^d$, where $V(x)\geq 0$, $V\in L^2_{\rm loc} (\mathbb R^d)$. We give a short proof of dimension free $L^p(\mathbb R^d)$ estimates, $1<p\leq 2$, for the vector of the Riesz…

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of…

We study the two dimensional Schr\"odinger operator, $H=-\Delta+V$, in the weighted L^1(\R^2) \rightarrow L^{\infty}(\R^2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\les \la x…

The spectrum of discrete Schr\"odinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the…

We study the cut-off resolvent of semiclassical Schr{\"o}dinger operators on $\mathbb{R}^d$ with bounded compactly supported potentials $V$. We prove that for real energies $\lambda^2$ in a compact interval in $\mathbb{R}_+$ and for any…

We derive a representation formula for the Weyl solution to the Schr\"odinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with…

In this paper, we study the Schr\"odinger equation associated with the Weinstein operators and we prove the existence and uniqueness of global solutions to Schr\"odinger-Weinstein equations in\\ $C\left(\left(-T_{\min }, T_{\max }\right) ;…

We determine the $L^p$-spectrum of the Schr\"odinger operator with the inverted harmonic oscillator potential $V(x)=-x^2$ for $1 \leq p \leq \infty$.

Let $L$ be a Schr\"odinger operator of the form $L=-\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ In this article we will show that a function…

We study, in the semiclassical limit, the singularly perturbed nonlinear Schr\"odinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schr\"odinger operator with a magnetic…

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials…

We provide a new analytical approach to operator splitting for equations of the type $u_t=Au+B(u)$ where $A$ is a linear operator and $B$ is quadratic. A particular example is the Korteweg-de Vries (KdV) equation $u_t-u u_x+u_{xxx}=0$. We…

We study the $L^p$-theory for the Schr\"odinger operator $\mathcal L_a$ with inverse-square potential $a|x|^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $(\mathcal L_a)^{s/2}$ agree with…

Let $(X,d,\mu)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincar\'{e} inequality. Let $\mathscr{L}=\mathcal{L}+V$ be a Schr\"odinger operator on $X$, where $\mathcal{L}$ is a non-negative operator…

The paper presents estimates for the number of negative eigenvalues of a two-dimensional Schr\"odinger operator in terms of $L\log L$ type Orlicz norms of the potential and proves a conjecture by N.N. Khuri, A. Martin and T.T. Wu.

In this paper, we introduce and investigate the fractional logarithmic $p$-Laplacian $(-\Delta)_{p}^{s+\log}$, defined as the first-order derivative with respect to the parameter $t$ of the fractional $p$-Laplacian $(-\Delta)_{p}^{t}$…

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schr\"odinger operator $H=-\Delta+V$ with a confining potential on $\mathbb{R}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the…