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Register a new userLifshits Tails for Squared Potentials
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We consider Schrodinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides.
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We consider Schr{o}dinger operators on $L^{2}({mathbb R}^{d})otimes L^{2}({mathbb R}^{ell})$ of the form $ H_{omega}~=~H_{perp}otimes I_{parallel} + I_{perp} otimes {H_parallel} + V_{omega}$, where $H_{perp}$ and $H_{parallel}$ are Schr{o}dinger operators on $L^{2}({mathbb R}^{d})$ and $L^{2}({mathbb R}^{ell})$ respectively, and $ V_omega(x,y)$ : = $sum_{xi in {mathbb Z}^{d}} lambda_xi(omega) v(x - xi, y)$, $x in {mathbb R}^d$, $y in {mathbb R}^ell$, is a random surface potential. We investigate the behavior of the integrated density of surface states of $H_{omega}$ near the bottom of the spectrum and near internal band edges. The main result of the current paper is that, under suitable assumptions, the behavior of the integrated density of surface states of $H_{omega}$ can be read off from the integrated density of states of a reduced Hamiltonian $H_{perp}+W_{omega}$ where $W_{omega}$ is a quantum mechanical average of $V_{omega}$ with respect to $y in {mathbb R}^ell$. We are particularly interested in cases when $H_{perp}$ is a magnetic Schr{o}dinger operator, but we also recover some of the results from [24] for non-magnetic $H_{perp}$.
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.
We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.
We prove Anderson localization at the internal band-edges for periodic magnetic Schr{o}dinger operators perturbed by random vector potentials of Anderson-type. This is achieved by combining new results on the Lifshitz tails behavior of the integrated density of states for random magnetic Schr{o}dinger operators, thereby providing the initial length-scale estimate, and a Wegner estimate, for such models.
We propose a solution method for studying relativistic spin-$0$ particles. We adopt the Feshbach-Villars formalism of the Klein-Gordon equation and express the formalism in an integral equation form. The integral equation is represented in the Coulomb-Sturmian basis. The corresponding Greens operator with Coulomb and linear confinement potential can be calculated as a matrix continued fraction. We consider Coulomb plus short range vector potential for bound and resonant states and linear confining scalar potentials for bound states. The continued fraction is naturally divergent at resonant state energies, but we made it convergent by an appropriate analytic continuation.
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