Dinger Operator Research Articles

A law of large numbers and a central limit theorem for the Schr�dinger operator with zero-range potentials

We consider the Schrodinger operator with zero-range potentials onN points of three-dimensional space, independently chosen according to a common distributionV(x). Under some assumptions we prove that, whenN goes to infinity, the sequence converges to a Schrodinger operator with an effective potential. The fluctuations around the limit operator are explicitly characterized.

Spectral theory of Schr�dinger operators with periodic complex-valued potentials

Spectral theory of Schr�dinger operators with periodic complex-valued potentials

Nonsingular finite-zone two-dimensional Schr�dinger operators and prymians of real curves

Nonsingular finite-zone two-dimensional Schr�dinger operators and prymians of real curves

New estimates on the number of bound states of Schr�dinger operators

In this Letter we prove inequalities on the number of bound states of Schrodinger operators by improving the method of Li and Yau relying on heat kernel estimates by Sobolev inequalities. We extend the range of applications and obtain better estimates. In particular, our technique yields a simple proof of Bargmann's bound and very good numerical values.

Bounds on the density of states of random Schr�dinger operators

Bounds are obtained on the unintegrated density of states ρ(E) of random Schrodinger operatorsH=−Δ + V acting onL 2(ℝ d ) orl 2(ℤ 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∈ℤ d . In the finite-difference case Λ(y) denotes the sitey∈ℤ d itself. Under the assumptionf ∈ L 0 1+ɛ (ℝ) it is proven that in the finitedifference casep ∈ L ∞(ℝ), and that in thed= 1 continuum casep ∈ L loc ∞ (ℝ).

Scattering for the Schr�dinger operator in the case of a crystal film

The elastic scattering (in the stationary and time-dependent approaches) for the Schroedinger operator with potential corresponding to a plane crystal film is described. The authors prove the existence of solutions of the Bloch analog of the Lippmann-Schwinger equation (stationary scattering states). They investigate the asymptotic behavior of these solutions and then the asymptotic behavior of the solutions of the time-dependent Schroedinger equation constructed by means of them. They obtain expressions for the probability of scattering of a particle near each of the preferred directions of reflection or transmission.

Note to the paper by R. Hempel, A. M. Hinz, and H. Kalf: On the essential spectrum of Schr�dinger operators with spherically symmetric potentials

Note to the paper by R. Hempel, A. M. Hinz, and H. Kalf: On the essential spectrum of Schr�dinger operators with spherically symmetric potentials

Treves inequality and the absence of positive eigenvalues for the Schr�dinger operator with a complex potential

One proves the absence of positive eigenvalues for the Schrodinger and Stark operators with the use of Hardy-type inequalities.

The Schr�dinger operator with a weakly accelerating potential

The ideas of scattering theory are applied to the construction of a unitary operator realizing the similarity of the operator - id/dξ in L2(ℝ) with a one-dimensional Schrodinger operator on the semiaxis with potential v(x), admitting at infinity the estimate .

Anderson localization for one-dimensional difference Schr�dinger operator with quasiperiodic potential

The Schrodinger difference operator considered here has the form $$(H_\varepsilon (\alpha )\psi )(n) = - (\psi (n + 1) + \psi (n - 1)) + V(n\omega + \alpha )\psi (n)$$ whereV is aC2-periodic Morse function taking each value at not more than two points. It is shown that for sufficiently smallɛ the operatorHɛ(α) has for a.e.α a pure point spectrum. The corresponding eigenfunctions decay exponentially outside a finite set. The integrated density of states is an incomplete devil's staircase with infinitely many flat pieces.

Localization for off-diagonal disorder and for continuous Schr�dinger operators

We extend the proof of localization by Delyon, Levy, and Souillard to accommodate the Anderson model with off-diagonal disorder and the continuous Schrodinger equation with a random potential.

A sufficient condition for the continuity of the spectrum of a discrete Schr�dinger operator

A sufficient condition for the continuity of the spectrum of a discrete Schr�dinger operator

Trace formula for Schr�dinger operator with Coulomb potential in three-dimensional space

A study is made of the Schroedinger operator in R/sup 3/ with potential equal to the sum of a Coulomb part and a rapidly decreasing part. For this operator, zeroth-order trace formulas are obtained that relate the determinant of the regularized scattering operator at zero energy to the characteristics of the discrete spectrum.

The infinite-dimensional Schr�dinger operator and its potential perturbations

The infinite-dimensional Schr�dinger operator and its potential perturbations

On the absence of resonances for Schr�dinger operators with non-trapping potentials in the classical limit

We provide bounds on resolvents of dilated Schrodinger operators via exterior scaling. This depends crucially on a non-trapping condition on the potential which has a clear interpretation in classical mechanics. These bounds imply absence of resonances due to the tail of the potential in the shape resonance problem.

Absence of localization in a class of Schr�dinger operators with quasiperiodic potential

We prove that a class of discrete Schrodinger operators with a quasi- periodic potential taking only a finite number of values, exhibits purely continuous spectrum; in particular they cannot have localized eigenvectors.

Asymptotics of the cut discontinuity and large order behaviour from the instanton singularity: The case of lattice Schr�dinger operators with exponential disorder

We discuss the relation between the singularity structure of the Borel transform, the asymptotics of the cut discontinuity and the large order behaviour of perturbation theory. In an explicit example — a tight binding model with exponential disorder — we show how to obtain the first instanton singularity from a cluster expansion in the Borel variable, and as an application we determine the exact decay of the density of states asE → ∞. The method opens some perspectives for similar problems arising from different models of Mathematical Physics.

Analog of Levinson's formula for a Schr�dinger operator with long-range potential

Analog of Levinson's formula for a Schr�dinger operator with long-range potential

Internal Lifschitz singularities of disordered finite-difference Schr�dinger operators

The integrated density of states has C∞-like singularities, ln|k(E)−k(Ec)|=−|E−Ec|−v/2 ϕc(E), with ϕc>0, a milder function at the edges of the spectral gaps which appear when the distribution function of the potentialdμ has a sufficiently large gap. The behaviour of ϕc nearEc is determined by the local continuity properties ofdμ near the relevant edge: ϕc(E)=O(1) ifdμ has an atom and ϕ=O(ln|E−Ec|) if μ is (absolutely) continuous and power bounded.

Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schr�dinger operator with a complex potential

The operator −(d/dx)2+ν where ν is a complex function, |J(x)(1+|x|1+e|>∞, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator −(d/dx)z−λ,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator −Δ+ν.