Related papers: Spectral determinant on graphs with generalized bo…

Commutator relations are used to investigate the spectra of Schr\"odinger Hamiltonians, $H = -\Delta + V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^{\nu}, \nu \geq d+1$. Here $\Delta$…

We consider the Schr\"odinger operator in the plane with delta-potential supported by a curve. For the cases of an infinite curve and a finite loop we give estimates on the lower bound of the spectrum expressed explicitly through the…

Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities…

In this article we consider the spectrum of a Laplacian matrix, also known as the Markov matrix, under the independence assumption. We assume that the entries have a variance profile. Motivated by recent works on generalized Wigner matrices…

A number of topics in the qualitative spectral analysis of the Schr\"odinger operator $-\Delta + V$ are surveyed. In particular, some old and new results concerning the positivity and semiboundedness of this operator as well as the…

We consider Sturm-Liouville operators on the line segment [0, 1] with general regular singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-…

The $p$-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex $p$-Laplacian and a hyperedge…

Boundedness of wave operators for Schr\"odinger operators in one space dimension for a class of singular potentials, admitting finitely many Dirac delta distributions, is proved. Applications are presented to, for example, dispersive…

We consider the Schr\"odinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$.…

We consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of…

We consider Schr\"odinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is…

In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space…

We study a general transition operator, generated by a random walk on a graph $X$; in particular we give necessary and sufficient condition on the matrix coefficient (1-step transition probablilities) to be a bounded operator from…

Let $M$ be a Riemannian manifold, $\tau: G \times M \to M$ an isometric action on $M$ of an $n$-torus $G$ and $V: M \to \mathbb R$ a bounded $G$-invariant smooth function. By $G$-invariance the Schr\"odinger operator, $P=-\hbar^2…

In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr\"odinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.

We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann boundary condition…

The construction of "sparse potentials", suggested in \cite{RS09} for the lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schr\"odinger operator $-\D-\a…

The periodic Schrodinger operator $ H $ on a discrete periodic graph is considered. We estimate the discrete spectrum of the perturbed operator $ H _ {-} (t) = H-tV $, $ t> 0 $, where the potential $ V \ ge 0 $ is decreasing and $t>0$ is…

A graph $G = (V, E)$ of bounded degree has an adjacency operator~$A$ which acts on the Hilbert space $\ell^2(V)$. There are different kinds of measures of interest on the spectrum $\Sigma (A)$ of $A$. In particular, each vector $\xi \in…

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative…