Positive Lyapunov exponents for a class of ergodic Schrödinger operators

We present a simple method to estimate the Lyapunov exponent (E) for the system

Localization for random Schrödinger operators with correlated potentials

We prove localization at high disorder or low energy for lattice Schrödinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance functionC(x,y) decays as |x–y|^–, where >0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.Partially supported by the NSF under grant PHY8515288Partially supported by the NSF under grant DMS8905627     

Lyapunov exponents of the Schrödinger equation with quasi-periodic potential on a strip

We prove that all the non-negative Lyapunov exponents of difference Schrödinger equation

One-dimensional Schrödinger operators with random decaying potentials

We investigate the spectrum of the following random Schrödinger operators:

Low energy asymptotics for Schrödinger operators with slowly decreasing potentials

Low energy behavior of Schrödinger operators with potentials which decay slowly at infinity is studied. It is shown that if the potential is positive then the zero energy is very regular and the resolvent is smooth near 0. This implies rapid local decay for the solutions of the Schrödinger equation. On the other hand, if the potential is negative then the resolvent has discontinuity at zero energy. Thus one cannot expect local decay faster than ordert ^–1 ast.     

Localization for a class of one dimensional quasi-periodic Schrödinger operators

We prove for small and satisfying a certain Diophantine condition the operator

Spectral properties of random Schrödinger operators with unbounded potentials

We investigate spectral properties of random Schrödinger operators H = - + _n()(1 + |n|) acting onl ²(Z ^d), where _n are independent random variables uniformly distributed on [0, 1].Research partially supported by a Sloan Doctoral Dissertation Fellowship and NSERC under grant OGP-0007901Research partially supported by NSF grant DMS-9101716     

Trace formulas for Schrödinger operators with complex potentials on a half line

Letters in Mathematical Physics - We consider Schrödinger operators with complex-valued decaying potentials on the half line. Such operator has essential spectrum on the half line plus...     

Inverse spectral problem for the 2-sphere Schrödinger operators with zonal potentials

Two basic problems of spectral theory of Schrödinger operators H=–+V(x) on the 2-sphere S ₂ are studied: (Direct problem) calculate large-k asymptotics of eigenvalue clusters {_kj}_j in terms of the potential function V; (Inverse problem) recover V from asymptotics of eigenvalue clusters. We get an explicit solution of the inverse problem and establish local spectral rigidity for zonal potentials V.The research was partly supported by the US NSF Grant DMS-8620231 and the Case Research Initiation Grant.     

A semi-classical trace formula for Schrödinger operators

LetS _?=??Δ+V, withV smooth. If 0<E ²V(x), the spectrum ofS _? nearE ² consists (for ? small) of finitely-many eigenvalues,λ _j (?). We study the asymptotic distribution of these eigenvalues aboutE ² as ?→0; we obtain semi-classical asymptotics for $$\sum\limits_j {f\left( {\frac{{\sqrt {\lambda _j (\hbar )} - E}}{\hbar }} \right)} $$ with \(\hat f \in C_0^\infty \) , in terms of the periodic classical trajectories on the energy surface \(B_E = \left\{ {\left| \xi \right|^2 + V(x) = E^2 } \right\}\) . This in turn gives Weyl-type estimates for the counting function \(\# \left\{ {j;\left| {\sqrt {\lambda _j (\hbar )} - E} \right| \leqq c\hbar } \right\}\) . We make a detailed analysis of the case when the flow onB _E is periodic.     

Singular continuous spectrum for palindromic Schrödinger operators

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is azX that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allxX ifX derives from a primitive substitution. For potentials defined by circle maps,x _n =1 _J (₀+n), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational and every half-open intervalJ.Work partially supported by NSERC.This material is based upon work supported by the National Science Foundation under Grant No. DMS-91-1715. The Government has certain rights in this material.     

Almost periodic Schrödinger operators

We studyH=–d ²/dx ²+V(x) withV(x) limit periodic, e.g.V(x)=a _n cos(x/2 ⁿ ) with a _n <. We prove that for a genericV (and for generica _n in the explicit example), (H) is a Cantor ( nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous.Research partially supported by NSF Grant MCS78-01885On leave from Department of Physics, Princeton UniversityOn leave from Departments of Mathematics and Physics, Princeton University; during 1980–81 Sherman Fairchild Visiting Scholar at Caltech     

Localization for some continuous random Schrödinger operators

We study the spectrum of random Schrödinger operators acting onL ²(R ^d ) of the following type . The are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.

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Résumé Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL ²(R ^d ) du type suivant . Les sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.

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U.R.A. 760 C.N.R.S.     

Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials

Consider the Schrödinger equation –u+V(x)u=u on the intervalI, whereV(x)0 forxI and where Dirichlet boundary conditions are imposed at the endpoints ofI. We prove the optimal bound

Internal Lifschitz singularities for one dimensional Schrödinger operators

The integrated density of states of the periodic plus random one-dimensional Schrödinger operator ;f0,q _i ()0, has Lifschitz singularities at the edges of the gaps inSp(H ). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.     

Decomposition of many-body Schrödinger operators

The complex-dilated many-body Schrödinger operatorH(z) is decomposed on invariant subspaces associated with the cuts {+z ^–2 R ⁺}, where is any threshold, and isolated spectral points. The interactions are dilation-analytic multiplicative two-body potentials, decaying asr ^–1+ atr=0 and asr ^–1+ atr=.