Related papers: Hill's potentials in H\"ormander spaces and their …

We analyze Schr\"odinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we…

Let $H$ be a quasiperiodic Schr\"{o}dinger operator generated by a monotone potential, as defined in [16]. Following [20], we study the connection between the Lyapunov exponent $L\left(E\right)$, arithmetic properties of the frequency…

The spectral properties of the Schr\"odinger operator $T_ty= -y''+q_ty$ in $L^2(\R)$ are studied, with a potential $q_t(x)=p_1(x), x<0, $ and $q_t(x)=p(x+t), x>0, $ where $p_1, p$ are periodic potentials and $t\in \R$ is a parameter of…

HVZ type theorem for semi-relativistic Pauli-Fierz Hamiltonian, $$\HHH=\sqrt{(p\otimes \one -A)^2+M^2}+V\otimes \one +\one\otimes \hf,\quad M\geq 0,$$ in quantum electrodynamics is studied. Here $H$ is a self-adjoint operator in Hilbert…

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…

We study the spectral properties of a system of electrons interacting through long-range Coulomb potential on a one-dimensional chain. When the interactions dominate over the electronic bandwidth, the charges arrange in an ordered…

We calculate the one-loop effective potential of a scalar field in a Robertson-Walker background with scalar metric perturbations. A complete set of orthonormal solutions of the perturbed equations is obtained by using the adiabatic…

Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,\pi].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large…

In space dimension $n\geq3$, we consider the electromagnetic Schr\"odinger Hamiltonian $H=(\nabla-iA(x))^2+V$ and the corresponding Helmholtz equation (\nabla-iA(x))^2u+u+V(x)u=f\quad \text{in}\quad \mathbb{R}^n, where the magnetic and…

We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous…

The one-dimensional Dirac operator \begin{equation*} L = i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{d}{dx} +\begin{pmatrix} 0 & P(x) \\ Q(x) & 0 \end{pmatrix}, \quad P,Q \in L^2 ([0,\pi]), \end{equation*} considered on $[0,\pi]$…

We study weak-coupling perturbation expansions for the ground-state energy of the Hamiltonian with the generalized spiked harmonic oscillator potential V(x) = Bx^2 + A/x^2 + lambda/x^alpha, and also for the bottoms of the angular momentum…

Let $\alpha\in(0,1)$ be an irrational, and $[0;a_1,a_2,...]$ the continued fraction expansion of $\alpha$. Let $H_{\alpha,V}$ be the one-dimensional Schr\"odinger operator with Sturm potential of frequency $\alpha$. Suppose the potential…

We study the spectral inequalities of Schr\"odinger operator in the whole space for different potentials, which can be power growth or continuously vanishing at infinity. The spectral inequalities quantitatively depend on the density of the…

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha, with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay as…

We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schr\"odinger operator whose spectra agree up to $O(h^\infty)$, yet their corresponding eigenvalues differ no less than exponentially. This result was…

Let $Q(x)$ denote a periodic function on the real line. The Schr\"odinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb{R})-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we…

We consider the Schr\"odinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics…

We investigate scattering, localization and dispersive time-decay properties for the one-dimensional Schr\"odinger equation with a rapidly oscillating and spatially localized potential, $q_\epsilon=q(x,x/\epsilon)$, where $q(x,y)$ is…

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr\"odinger operator ${\bf H}$ with such a potential.…