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=.     

Super-extensions of energy dependent Schrödinger operators

We consider the energy dependent super Schrödinger linear problem which is a direct generalization of the purely even, energy dependent Schrödinger equation discussed in [1]. We show that the isospectral flows of that problem possess (N+1) compatible Hamiltonian structures. We also extend a generalised factorisation approach of [2] to this case and derive a sequence ofN modifications for the 2N component systems. Then ^th such modification possesses (N–n+1) compatible Hamiltonian structures.On leave of absence from Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-00-681 Warsaw, Poland (present address)     

On the spectra of Schrödinger operators

We give two formulas for the lowest point in the spectrum of the Schrödinger operatorL=–(d/dt)p(d/dt)+q, where the coefficientsp andq are real-valued, bounded, uniformly continuous functions on the real line. We determine whether or not is an eigenvalue forL in terms of a set of probability measures on the maximal ideal space of theC ^*-algebra generated by the translations ofp andq.Research supported in part by the National Science Foundation under Grant DMS-910496     

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     

Schrödinger operators withL loc p -potentials

We discuss the question of when the closure of the Schrödinger operator, –+V, acting inL ^p(R ^l,d ^lx), generates a strongly continuous contraction semigroup. We prove a series of theorems proving the stability for –:L ^pL ^p of the property of having am-accretive closure under perturbations by functions inL _loc ^q (1<pq). The connection with form sums and the Trotter product formula are considered. These results generalize earlier results of Kato, Kalf-Walter, Semenov and Beliy-Semenov in that we allow more general local singularities, including arbitrary singularities at one point, and arbitrary growth at infinity. We exploit bilinear form methods, Kato's inequality and certain properties of infinitesimal generators of contractions.     

Bounds on the density of states of random Schrödinger operators

Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operatorsH=?Δ + V acting onL ²(? ^d ) orl ²(? ^d ). In both cases the random potential is $$V: = \sum\limits_{y \in \mathbb{Z}^d } {V_y \chi (\Lambda (y))}$$ in which the \(\left\{ {V_y } \right\}_{y \in \mathbb{Z}^d }\) areIID random variables with densityf. The χ denotes indicator function, and in the continuum case the \(\left\{ {\Lambda (y)} \right\}_{y \in \mathbb{Z}^d }\) are cells of unit dimensions centered ony∈? ^d . In the finite-difference case Λ(y) denotes the sitey∈? ^d itself. Under the assumptionf ∈ L ₀ ^1+? (?) it is proven that in the finitedifference casep ∈ L ^∞(?), and that in thed= 1 continuum casep ∈ L _loc ^∞ (?).     

Antibound states for a class of one-dimensional Schrödinger operators

Let (x) be the Dirac's delta,q(x)L ¹ (R)L ² (R) be a real valued function, and , R; we will consider the following class of one-dimensional formal Schrödinger operators on . It is known that to the formal operator may be associated a selfadjoint operatorH(,) onL ²(R). Ifq is of finite range, for >0 and || is small enough, we prove thatH(,) has an antibound state; that is the resolvent ofH(,) has a pole on the negative real axis on the second Riemann sheet.Work done while the author was supported by an undergraduate fellowship of the (Italian) National Research Council (CNR).     

On stochastic methods in spectral theory of Schrödinger operators

The Feynman-Kac formula for regular and singular potentials is used to investigate spectral properties of hard-core type Schrödinger operators. The stability of the essential and absolutely continuous spectra is described. Trace class and Hilbert-Schmidt class properties for the corresponding semigroups are considered.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

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     

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.     

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.     

Non-standard finite-difference time-domain method for solving the Schrödinger equation

The literature on chaos has highlighted several chaotic systems with special features. In this work, a novel chaotic jerk system with non-hyperbolic equilibrium is proposed. The dynamics of this new system is revealed through equilibrium analysis, phase portrait, bifurcation diagram and Lyapunov exponents. In addition, we investigate the time-delay effects on the proposed system. Realisation of such a system is presented to verify its feasibility.     

One-dimensional Schrödinger operators with random decaying potentials

We investigate the spectrum of the following random Schrödinger operators:

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.     

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

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

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

Positive Lyapunov exponents for a class of ergodic Schrödinger operators

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

On the spectrum of Schrödinger operators with a random potential

We investigate the spectrum of Schrödinger operatorsH of the type:H =–+q _i ()f(x–x _i + _i ())(q _i () and _i () independent identically distributed random variables,i ^d ). We establish a strong connection between the spectrum ofH and the spectra of deterministic periodic Schrödinger operators. From this we derive a condition for the existence of forbidden zones in the spectrum ofH . For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly.