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Register a new userStark resonances in 2-dimensional curved quantum waveguides
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In this paper we study the influence of an electric field on a two dimen-sional waveguide. We show that bound states that occur under a geometrical deformation of the guide turn into resonances when we apply an electric field of small intensity having a nonzero component on the longitudinal direction of the system. MSC-2010 number: 35B34,35P25, 81Q10, 82D77.
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We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide ${bf Omega}$ i.e. bound states of the corresponding Laplace operator $-Delta_{{bf Omega}}$. Here the curvature of the guide is supposed to satisfy some assumptions of analyticity, and decays as $O(|s|^{-varepsilon}), varepsilon > 3$ at infinity. We show that under conditions on the electric field $ bf F$, ${bf H}(F):= -Delta_{{bf Omega}} + {bf F}. {bf x} $ has resonances near the discrete eigenvalues of $-Delta_{{bf Omega}}$.
We define resonances for finitely perturbed quantum walks as poles of the scattering matrix in the lower half plane. We show a resonance expansion which describes the time evolution in terms of resonances and corresponding Jordan chains. In particular, the decay rate of the survival probability is given by the imaginary part of resonances and the multiplicity. We prove generic simplicity of the resonances, although there are quantum walks with multiple resonances.
We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist.
The question of whether it is possible to compute scattering resonances of Schrodinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be $mathcal{C}^1$. The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested point.
We consider the Dirichlet Laplacian $H_gamma$ on a 3D twisted waveguide with random Anderson-type twisting $gamma$. We introduce the integrated density of states $N_gamma$ for the operator $H_gamma$, and investigate the Lifshits tails of $N_gamma$, i.e. the asymptotic behavior of $N_gamma(E)$ as $E downarrow inf {rm supp}, dN_gamma$. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.
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