Related papers: Eigenvalue bounds for Schr\"odinger operators on q…
We establish quantitative upper and lower bounds for Schr\"odinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys.,…
We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…
In this paper, we discuss the concept of quantum graphs with transparent vertices by considering the case where the graph interacts with an external time-independent field. In particular, we address the problem of transparent boundary…
This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is…
We prove dynamical upper bounds for discrete one-dimensional Schroedinger operators in terms of various spacing properties of the eigenvalues of finite volume approximations. We demonstrate the applicability of our approach by a study of…
We consider a Schr\"odinger particle on a graph consisting of $\,N\,$ links joined at a single point. Each link supports a real locally integrable potential $\,V_j\,$; the self--adjointness is ensured by the $\,\delta\,$ type boundary…
We make a spectral analysis of discrete Schroedinger operators on the half-line, subject to complex Robin-type boundary couplings and complex-valued potentials. First, optimal spectral enclosures are obtained for summable potentials.…
We consider random Schr\"odinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to…
We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the…
We study Schr\"{o}dinger operators on star metric graphs with potentials of the form $\alpha\varepsilon^{-2}Q(\varepsilon^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions…
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…
We consider a class of quasilinear operators on a bounded domain $\Omega\subset \mathbb R^n$ and address the question of optimizing the first eigenvalue with respect to the boundary conditions, which are of the Robin-type. We describe the…
We consider the Schr\"odinger equation for hydrogen-like atom with Coulomb potential and non-point ball nucleus. The eigenvalues and eigenfunctions of the operator given by an arbitrary rotation-invariant boundary value problem on the…
We consider periodic Schr\"{o}dinger operators on the hexagonal lattice with self-adjoint vertex conditions that allow discontinuity and concentrated mass at the vertices. This model generalizes the periodic Schr\"{o}dinger operator on the…
Norm resolvent approximation for a wide class of point interactions in one dimension is constructed. To analyse the limit behaviour of Schr\"odinger operators with localized singular rank-two perturbations coupled with {\delta}-like…
We extend the surgical tool box for quantum graphs to anti-standard and $\delta'$ vertex conditions. Monotonicity properties of eigenvalues of graph Laplacian with $\delta'$ interactions at vertices depend on the sign of vertex parameter.…
We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and…
This paper is concerned with the estimation of the number of negative eigenvalues (bound states) of Schroedinger operators in a strip subject to Neumann boundary conditions. The estimates involve weighted L^1 norms and L ln L norms of the…
We prove Lieb-Thirring inequalities for Schr\"odinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength…
We derive a sharp bound on the location of non-positive eigenvalues of Schroedinger operators on the halfline with complex-valued potentials.