Related papers: On Absence of Threshold Resonances for Schrodinger…

In this paper we study on $L^2(\mathbb{R}^d)$ the quasi-periodic Schr\"odinger operator $H=-\Delta+ \lambda V(x),$ where $V$ is a real analytic quasi-periodic function and $\lambda>0$. We first show that $H$ has no eigenvalues in…

In this talk, we report on results about the width of the resonances for a slowly varying perturbation of a periodic operator. The study takes place in dimension one. The perturbation is assumed to be analytic and local in the sense that it…

We consider the Schr\"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the…

By the Aharonov-Casher theorem, the Pauli operator $P$ has no zero eigenvalue when the normalized magnetic flux $\alpha$ satisfies $|\alpha|<1$, but it does have a zero energy resonance. We prove that in this case a Lieb-Thirring inequality…

For any pseudo-Riemannian hyperbolic space $X$ over $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$, we show that the resolvent $R(z)=(\Box-z\operatorname{Id})^{-1}$ of the Laplace-Beltrami operator $-\Box$ on $X$ can be extended…

We consider the large $L$ limit of one dimensional Schr\"odinger operators $H_L=-d^2/dx^2 + V_1(x) + V_{2,L}(x)$ in two cases: when $V_{2,L}(x)=V_2(x-L)$ and when $V_{2,L}(x)=e^{-cL}\delta(x-L)$. This is motivated by some recent work of…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable…

We show that additional solutions must be ignored (in differences of the Schrodinger and Klein-Gordon equations) in the Dirac equation, where usually passed the second order radial equation, called the reduced equation, instead of a system.…

A spectral theory of linear operators on a rigged Hilbert space is applied to Schr\"odinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach…

We consider a semiclassical $2\times 2$ matrix Schr\"odinger operator of the form $P=-h^2\Delta {\bf I}_2 + {\rm diag}(x_n-\mu, \tau V_2(x)) +hR(x,hD_x)$, where $\mu$ and $\tau$ are two small positive constants, $V_2$ is real-analytic and…

In the present article we show that the energy spectrum of the one-dimensional Dirac equation, in the presence of an attractive vectorial delta potential, exhibits a resonant behavior when one includes an asymptotically spatially vanishing…

For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \\ Q & 0 \end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, $$ subject to periodic…

We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed…

The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown…

We prove a limiting absorption principle at zero energy for two-body Schr\"odinger operators with long-range potentials having a positive virial at infinity. More precisely, we establish a complete asymptotic expansion of the resolvent in…

An 1D tight-binding version of the Dirac equation is considered; after checking that it recovers the usual discrete Schr?odinger equation in the nonrelativistic limit, it is found that for two-valued Bernoulli potentials the zero mass case…

We consider singular self-adjoint extensions for the Schr\"{o}dinger operator of spin-$1/2$ particle in one dimension. The corresponding boundary conditions at a singular point are obtained. There are boundary conditions with the spin-flip…

Eigenvalue behaviors of Schr\"odinger operator defined on $n$-dimensional lattice with $n+1$ delta potentials is studied. It can be shown that lower threshold eigenvalue and lower threshold resonance are appeared for $n\geq 2$, and lower…

We consider resonances for third order ordinary differential operator with compactly supported coefficients on the real line. Resonance are defined as zeros of a Fredholm determinant on a non-physical sheet of three sheeted Riemann surface.…

We characterize the potential V (x) that minimizes the fundamental spectral gap of weighted Schr\"odinger operators on the interval [0,{\pi}] subject to Dirichlet boundary conditions, under the constraint that the potential V (x) is convex…