Related papers: Schroedinger Operators on Regular Metric Trees wit…

In this paper we deal with the following weakly coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} - \Delta_\alpha u + \omega u = |u|^2 u + \beta u |v|^2&\quad \mathrm{in}\ \mathbb{R}^2,\\ - \Delta v + \tilde{\omega} v =…

The spectral properties of two-dimensional Schr\"odinger operators with $\delta'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete…

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 aim of this paper is to review and compare the spectral properties of (the closed extension of) --$\Delta$ + U (V $\ge$ 0) and --$\Delta$ + iV in L 2 (R^d) for C $\infty$ real potentials U or V with polynomial behavior. The case with…

The behavior of the discrete spectrum of the Schr\"odinger operator $-\D - V$, in quite a general setting, up to a large extent is determined by the behavior of the corresponding heat kernel $P(t;x,y)$ as $t\to 0$ and $t\to\infty$. If this…

In this paper, we are interested in the nonlinear Schr\"odinger problem $-\Delta u + Vu = \abs{u}^{p-2}u$ submitted to the Dirichlet boundary conditions. We consider $p>2$ and we are working with an open bounded domain $\Omega\subset\IR^N$…

Let P be the operator $-\Delta+V$ on R^d, where $V$ is a real potential with several inverse square singularities. The usual non-trapping type high-frequency inequality on the truncated resolvent of $P$ is shown, using semi-classical…

We investigate the spectrum of Schr\"odinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to $-\infty$, a narrow cluster of…

We establish sharp pointwise estimates for the ground states of some singular fractional Schr\"odinger operators on relatively compact Euclidean subsets. The considered operators are of the type $(-\Delta)^{\alpha/2}|_\Omega-V$, where $V\in…

We analyze properties of semigroups generated by Schr\"odinger operators $-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant…

Let $H_V=-\Delta +V$ be a Schr\"odinger operator on an arbitrary open set $\Omega$ of $\mathbb R^d$, where $d \geq 3$, and $\Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $\Omega$. The purpose of this…

We analyse the spectral phase diagram of Schr\"odinger operators $ T +\lambda V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by iid random variables. The main result is a criterion for the…

In this paper we consider the three-dimensional Schr\"{o}dinger operator with a $\delta$-interaction of strength $\alpha > 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,\beta f(x))$, where…

We shall consider the Schr\"odinger operators on $\mathbf{R}^2$ with random $\delta$ magnetic fields. Under some mild conditions on the positions and the fluxes of the $\delta$-fields, we prove the spectrum coincides with $[0,\infty)$ and…

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-\Delta)^{M/2}+\mathcal{V}\ , \quad M>0\ , \end{equation} on $L^2(\mathbb{R}^d/\Gamma)$, with $\Gamma$ a maximal dimension lattice in $\mathbb{R}^d$ and…

The subject of the paper are Schr\"odinger operators on tree graphs which are radial having the branching number $b_n$ at all the vertices at the distance $t_n$ from the root. We consider a family of coupling conditions at the vertices…

The lowest eigenvalue of the Schr\"odinger operator $-\Delta+\mathcal{V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.

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…

We study the Schr\"odinger operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\R^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a…

Schr\"{o}dinger operators of the form $\Delta - W$ on $L^2_{\text{rad}}(\mathbb{R}^3)$, the space of radially symmetric square integrable functions are relevant in a variety of physical contexts. The potential $W$ is taken to be radially…