One-dimensional Schrödinger operators with random decaying potentials

We investigate the spectrum of the following random Schrödinger operators:

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     

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.     

Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

Theq=0 combinatorics for is studied in connection with solvable lattice models. Crystal bases of highest weight representations of are labelled by paths which were introduced as labels of corner transfer matrix eigenvectors atq=0. It is shown that the crystal graphs for finite tensor products ofl-th symmetric tensor representations of approximate the crystal graphs of levell representations of . The identification is made between restricted paths for the RSOS models and highest weight vectors in the crystal graphs of tensor modules for .Partially supported by NSF grant MDA904-90-H-4039     

Discrete spectrum of Schrödinger operators with potentials concentrated near conical surfaces

Letters in Mathematical Physics - In this paper, we study spectral properties of a three-dimensional Schrödinger operator $$-Delta +V$$ with a potential V given, modulo rapidly decaying...     

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.

---

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.

---

---

U.R.A. 760 C.N.R.S.     

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 ^∞ (?).     

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.     

Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

Letters in Mathematical Physics - We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or...     

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=.     

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.     

Nonlinear Schrödinger equation with complex supersymmetric potentials

Using the concept of supersymmetry we obtain exact analytical solutions of nonlinear Schrödinger equation with a number of complex supersymmetric potentials and power law nonlinearity. Linear stability of these solutions for self-focusing as well as de-focusing nonlinearity has also been examined.     

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     

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)     

Spectral and scattering theory for a Schrödinger operator with a class of momentum-dependent long-range potentials

Let be the selfadjoint operator for the static electromagnetic field where W _j for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at as |x|^- with in (0, 1]. Then for >1/2, asymptotic completeness holds for the scattering system (H, H ₀).     

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     

Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials

In this paper, we study the Lyapunov stabilities of some semiclassical bound states of the (nonhomogeneous) nonlinear Schrödinger equation, We prove that among those bound states, those which are concentrated near local minima (respectively maxima) of the potentialV are stable (respectively unstable). We also prove that those bound states are positive if is sufficiently small.     

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