A Second Eigenvalue Bound for the Dirichlet Schrödinger Operator

Let λ _i (Ω,V) be the i ^th eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^nLet λ _i (Ω,V) be the i ^th eigenvalue of the Schr?dinger operator with Dirichlet boundary conditions on a bounded domain and with the positive potential V. Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V _*, we prove that λ₂(Ω,V) ≤ λ₂(S ₁,V _*). Here S ₁ denotes the ball, centered at the origin, that satisfies the condition λ₁(Ω,V)=λ₁(S ₁,V _*).Further we prove under the same convexity assumptions on a spherically symmetric potential V, that λ₂(B _R , V) / λ₁(B _R , V) decreases when the radius R of the ball B _R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.R.B. was supported by FONDECYT project # 102-0844.H.L. gratefully acknowledges financial support from DIPUC of the Pontifí cia Universidad Católica de Chile and from CONICYT.     

Dynamical Localization for the Random Dimer Schrödinger Operator

We study the one-dimensional random dimer model, with Hamiltonian H _ω =Δ+V _ω , where for all x∈ $\mathbb{Z}$ , V _ω(2x)=V _ω(2x+1) and where the V _ω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ $\sqrt 2$ , the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/ $\sqrt 2$ , respectively, V= $\sqrt 2$ , we show the existence of new additional critical energies at E=±3/ $\sqrt 2$ , respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ $\ell$ ²( $\mathbb{Z}$ ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here $\psi _t = e^{- iH_{\omega ^t}} \psi$ , and P _I(H _ω) is the spectral projector of H _ωonto the interval I. In particular, if V>1 and V≠ $\sqrt 2$ , these results hold on the entire spectrum [so that one can take I=σ(H _ω)].     

The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrödinger operators

$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$

, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω _k are independent random variables with mean 0 and variance 1.

We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.     

Lifshitz Tail for Schrödinger Operator¶with Random Magnetic Field

We study the behavior of the density states at the lower edge of the spectrum for Schr?dinger operators with random magnetic fields. We use a new estimate on magnetic Schr?dinger operators, which is similar to the Avron–Herbst–Simon estimate but the bound is always nonnegative. Received: 3 January 2000 / Accepted: 18 April 2000     

Asymptotic Lower Bounds for a Class of Schrödinger Equations

We shall study the following initial value problem:

L p -Boundedness of the Wave Operator for the One Dimensional Schrödinger Operator

Given a one dimensional perturbed Schrödinger operator H =  ? d ²/dx ² + V(x), we consider the associated wave operators W  _± , defined as the strong L ² limits $\lim_{s\to\pm\infty}e^{isH}e^{-isH_{0}}Given a one dimensional perturbed Schr?dinger operator H = − d ²/dx ² + V(x), we consider the associated wave operators W _± , defined as the strong L ² limits . We prove that W _± are bounded operators on L ^p for all 1 < p < ∞, provided , or else and 0 is not a resonance. For p = ∞ we obtain an estimate in terms of the Hilbert transform. Some applications to dispersive estimates for equations with variable rough coefficients are given.     

Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Given n2, we put r=min . Let be a compact, C ^r -smooth surface in ⁿ which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L ²( ⁿ ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.     

Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ? _N , the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as \({\frac{\pi^{2}}{(\ell_{N} +1)^{2}}} \) in the sense that the ratio of the quantities goes to one.     

Spectrum of the Schrödinger Operator in a Perturbed Periodically Twisted Tube

We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted tube of a non-circular cross section. It is shown that a local perturbation which consists of “slowing down” the twisting in the mean gives rise to a non-empty discrete spectrum Mathematical Subject Classifications: 35P05, 81Q10.     

Semiclassical Eigenfunctions of the Schrödinger Operator on a Graph That Are Localized Near a Subgraph

We study semiclassical eigenvalues and eigenfunctions of the Schrödinger operator on a geometric graph. We show that nontrivial boundary conditions at vertices lead to the existence of eigenfunctions, concentrated near a single vertex. We also construct semiclassical eigenfunctions, localized near edges and discuss general construction of spectral series which correspond to a general subgraph.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. III. Localization Properties

In this paper, we study the asymptotic localization properties with high probability of the Kth eigenfunction (associated with the Kth largest eigenvalue, K?1 fixed) of the multidimensional Anderson Hamiltonian in torus V increasing to the whole of lattice. Denote by z _K,V ∈V the site at which the Kth largest value of potential is attained. It is well-known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity, then the Kth eigenfunction is asymptotically delta-function at the site z _τ(K),V (localization centre) for some random τ(K)=τ _V (K)?1. We study the asymptotic behavior of the index τ _V (K) by distinguishing between three cases of the tails of potential distribution: (i) for the “heavy tails” (including Gaussian), τ _V (K) is asymptotically bounded; (ii) for the light tails, but heavier than the double exponential, the index τ _V (K) unboundedly increases like |V|^o(1); (iii) finally, for the double exponential tails with high disorder, the index τ _V (K) behaves like a power of |V|. For Weibull’s and fractional-double exponential types distributions associated with the case (ii), we obtain the first order expansion formulas for logτ _V (K).     

On Levels of a Weakly Perturbed Periodic Schrödinger Operator

We consider the Schrödinger operator with a periodic potential perturbed by a function which is periodic in two variables and exponentially decreases in third variable. When the perturbation is small it is proved that the levels (eigenvalues or resonances) exist near the stationary points of the eigenvalues of the periodic Schrödinger operator in the cell with respect to the third component of quasimomentum. The behaviour of these levels is investigated.     

Lifshitz Tails for a Class of Schrödinger Operators with Random Breather-Type Potential

We derive bounds on the integrated density of states for a class of Schrödinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrödinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.     

Symmetries of Schrödinger Operator with Point Interactions

The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling W are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.     

Long Time Anderson Localization for the Nonlinear Random Schrödinger Equation

We prove long time Anderson localization for the nonlinear random Schrödinger equation for arbitrary ? ² initial data, hence giving an answer to a widely debated question in the physics community. The proof uses a Birkhoff normal form type transform to create a barrier where there is essentially no propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat “rough” data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.     

On the Bound States for the Three-Body Schrödinger Equation with Decaying Potentials

The three-body Schrödinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length ${\kappa + 1, \kappa = 0, 1, \ldots}$ As a result, the solutions to the three-body Schrödinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrödinger equation in three dimensions with the inverse power potential of the form ${r^{-{\kappa}-1}}$ . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with ${\kappa = 0}$ , are reduced to the well-known eigenvalues of bound states at threshold.     

Bound state solutions to the Schrödinger equation for some diatomic molecules

The radiation damage in the structural materials of a 500 MWe Indian prototype fast breeder reactor (PFBR) is re-assessed by computing the neutron displacement per atom (dpa) cross-sections from the recent nuclear data library evaluated by the USA, ENDF / B-VII.1, wherein revisions were taken place in the new evaluations of basic nuclear data because of using the state-of-the-art neutron cross-section experiments, nuclear model-based predictions and modern data evaluation techniques. An indigenous computer code, computation of radiation damage (CRaD), is developed at our centre to compute primary-knock-on atom (PKA) spectra and displacement cross-sections of materials both in point-wise and any chosen group structure from the evaluated nuclear data libraries. The new radiation damage model, athermal recombination-corrected displacement per atom (arc-dpa), developed based on molecular dynamics simulations is also incorporated in our study. This work is the result of our earlier initiatives to overcome some of the limitations experienced while using codes like RECOIL, SPECTER and NJOY 2016, to estimate radiation damage. Agreement of CRaD results with other codes and ASTM standard for Fe dpa cross-section is found good. The present estimate of total dpa in D-9 steel of PFBR necessitates renormalisation of experimental correlations of dpa and radiation damage to ensure consistency of damage prediction with ENDF / B-VII.1 library.