Dinger Operator Research Articles

Schrödinger operator on the axis with potentials depending on two parameters

A Schrödinger operator on the axis is considered; its localized potential is the sum of a small potential and certain potentials with contracting supports, which can increase unboundedly when their supports are contracted. Sufficient conditions are presented for the absence (or existence) of eigenvalues for such an operator. In the case where eigenvalues exist, their asymptotic expansion is constructed.

Magnetic Fourier integral operators

In some previous papers we have defined and studied a 'magnetic' pseudodifferential calculus as a gauge covariant generalization of the Weyl calculus when a magnetic field is present. In this paper we extend the standard Fourier Integral Operators Theory to the case with a magnetic field, proving composition theorems, continuity theorems in 'magnetic' Sobolev spaces and Egorov type theorems. The main application is the representation of the evolution group generated by a 1-st order 'magnetic' pseudodifferential operator (in particular the relativistic Schr\"{o}dinger operator with magnetic field) as such a 'magnetic' Fourier Integral Operator. As a consequence of this representation we obtain some estimations for the distribution kernel of this evolution group and a result on the propagation of singularities.

Poisson statistics for eigenvalues of continuum random Schrödinger operators

We show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the local- ization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point pro- cess with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues. We derive a Minami estimate for continuum An- derson Hamiltonians. We also give a simple and transparent proof of Minami's estimate for the (discrete) Anderson model. In this article we show absence of energy levels repulsion for the eigenvalues of random Schrodinger operators in the continuum. We prove that, in the localiza- tion region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We also obtain simplicity of the eigenvalues in that region. Local fluctuations of eigenvalues of random operators is believed to distinguish between localized and delocalized regimes, indicating an Anderson metal-insulator transition. Exponential decay of eigenfunctions implies that disjoint regions of space are uncorrelated and create almost independent eigenvalues, and thus absence of energy levels repulsion, which is mathematically translated in terms of a Poisson point process. On the other hand, extended states imply that distant regions have mutual influence, and thus create some repulsion between energy levels. Local fluctuations of eigenvalues have been studied within the context of random matrix theory, in particular Wigner matrices and GUE matrices, cf. (B, DiPS, ESY1, ESY2, J1, J2, SS) and references therein. It is challenging to understand random hermitian band matrices from the perspective of their eigenvalues fluctuations, by proving a transition between Poisson statistics and a semi-circle law for the density of states (a signature of energy levels repulsion), and relate this to the (discrete) Anderson model, cf. (B, DiPS). CMV matrices are another class of random matrices for which Poisson statistics and a transition to energy levels repulsion have been proved (KS, St1, St2). For random Schrodinger operators, Poisson statistics for eigenvalues was first proved by Molchanov (Mo2) for the same one-dimensional continuum random Schro- dinger operator for which Anderson localization was first rigorously established

H1 BOUNDEDNESS FOR RIESZ TRANSFORM RELATED TO SCHRO¨ DINGER OPERATOR ON NILPOTENT GROUPS

Let $\gg$ be a nilpotent Lie groups equipped with a H\{o}rmander system of vector fields $X=(X_1,\ldots,X_m)$ and $\Delta=\sum_{i=1}^mX_i^2$ be the sub-Laplacians associated with $X$. Let $A=-\Delta +W$ be the Schr\{o}dinger operator with the potential function $W$ belongs to the reverse H\{o}lder class $B_q$ for some $q\ge D/2$, where $D$ denote the dimension at infinity. In this paper, we prove that the Riesz transform $\nabla A^{-1/2}$ related to Schr\{o}dinger operator $A$ is bounded from the Hardy space $H^1(\gg)$ to itself.

Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II

Let $\Omega \subset \mathbb {R}^N$ be a compact imbedded Riemannian manifold of dimension $d\ge 1$ and define the $(d+1)$-dimensional Riemannian manifold $\mathcal {M}:=\{(x,r(x)\omega )\::\: x\in \mathbb {R}, \omega \in \Omega \}$ with $r>0$ and smooth, and the natural metric $ds^2=(1+r’(x)^2)dx^2+r^2(x)ds_\Omega ^2$. We require that $\mathcal {M}$ has conical ends: $r(x)=|x| + O(x^{-1})$ as $x\to \pm \infty$. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution $e^{it\Delta _\mathcal {M}}$ and the wave evolution $e^{it\sqrt {-\Delta _\mathcal {M}}}$ are obtained for data of the form $f(x,\omega )=Y_n(\omega ) u(x)$, where $Y_n$ are eigenfunctions of $-\Delta _\Omega$ with eigenvalues $\mu _n^2$. In this paper we discuss all cases $d+n>1$. If $n\ne 0$ there is the following accelerated local decay estimate: with \[ 0< \sigma = \sqrt {2\mu _n^2+(d-1)^2/4}-\frac {d-1}{2} \] and all $t\ge 1$, \[ \Vert w_\sigma e^{it\Delta _{\mathcal {M}}} Y_nf \Vert _{L^{\infty }(\mathcal {M})} \le C(n,\mathcal {M},\sigma ) t^{-\frac {d+1}{2}-\sigma }\Vert w_\sigma ^{-1} f\Vert _{L^1(\mathcal {M})}, \] where $w_\sigma (x)=\langle x\rangle ^{-\sigma }$, and similarly for the wave evolution. Our method combines two main ingredients: (A) A detailed scattering analysis of Schrödinger operators of the form $-\partial _\xi ^2 + (\nu ^2-\frac 14)\langle \xi \rangle ^{-2}+U(\xi )$ on the line where $U$ is real-valued and smooth with $U^{(\ell )}(\xi )=O(\xi ^{-3-\ell })$ for all $\ell \ge 0$ as $\xi \to \pm \infty$ and $\nu >0$. In particular, we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the Wronskian between the outgoing Jost solutions as the energy tends to zero. In particular, the division into Part I and Part II can be explained by the former being resonant at zero energy, where the present paper deals with the nonresonant case. (B) Estimation of oscillatory integrals by (non)stationary phase.

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schro dinger operator) H = −d 2 /dx 2 + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrodinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion. The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.

On the Perturbation Theory of Self-Adjoint Operators

We show that all types of self-adjoint perturbations of a semi-bounded operator $A$ (purely singular, mixed singular, and regular) can be described and studied from a unique point of view in the framework of the extension theory as well as in the framework of the additive perturbation theory. We also show that any singular finite rank perturbation $\widetilde{A}$ can be approximated in the norm resolvent sense by regular finite rank perturbations of $A$. An application is given to the study of Schr\"{o}dinger operators with point interactions.

A finiteness theorem for the space of $L^{p}$ harmonic sections

In this paper we give a unified and improved treatment to finite dimensionality results for subspaces of L^{p} harmonic sections of Riemannian or Hermitian vector bundles over complete manifolds. The geometric conditions on the manifold are subsumed by the assumption that the Morse index of a related Schr#x00F6;dinger operator is finite. Applications of the finiteness theorem to concrete geometric situations are also presented.

Explicit invariant measures for products of random matrices

We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of $\text {SL}(2,{\mathbb C})$. The matrices in the product are such that one entry is gamma-distributed along a ray in the complex plane. When the ray is the positive real axis, the products are those associated with a continued fraction studied by Letac & Seshadri [Z. Wahr. Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the continued fraction is a generalised inverse Gaussian. We extend this result by finding the distribution for an arbitrary ray in the complex right-half plane, and thus compute the corresponding Lyapunov exponent explicitly. When the ray lies on the imaginary axis, the matrices in the infinite product coincide with the transfer matrices associated with a one-dimensional discrete Schrödinger operator with a random, gamma-distributed potential. Hence, the explicit knowledge of the Lyapunov exponent may be used to estimate the (exponential) rate of localisation of the eigenstates.

Semiclassical asymptotics and gaps in the spectra of periodic Schrödinger operators with magnetic wells

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold $M$ such that $H^1(M, \mathbb {R})=0$, equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries, has an arbitrarily large number of spectral gaps in the semi-classical limit.

Formula omitted] boundedness of commutators of Riesz transforms associated to Schrödinger operator

In this paper we consider L p boundedness of some commutators of Riesz transforms associated to Schrödinger operator P = − Δ + V ( x ) on R n , n ⩾ 3 . We assume that V ( x ) is non-zero, non-negative, and belongs to B q for some q ⩾ n / 2 . Let T 1 = ( − Δ + V ) −1 V , T 2 = ( − Δ + V ) − 1 / 2 V 1 / 2 and T 3 = ( − Δ + V ) − 1 / 2 ∇ . We obtain that [ b , T j ] ( j = 1 , 2 , 3 ) are bounded operators on L p ( R n ) when p ranges in a interval, where b ∈ BMO ( R n ) . Note that the kernel of T j ( j = 1 , 2 , 3 ) has no smoothness.

On the Two Spectra Inverse Problem for Semi-infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.

The essential spectrum of Schrödinger operators on lattices

The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schr\"{o}dinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schr\"{o}dinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schr\"{o}dinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schr\"{o}dinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.

Fredholmness of pseudodifference operators in weighted Spaces

The aim of the paper is to present some results concerning pseudodifference operators on ℤN, which are a discrete analog of standard pseudodifferential operators on ℝN. We study the Fredholm property of pseudodifference operators acting in weighted lp spaces on ℤN and the Phragmen-Lindelof principle for solutions of pseudodifference equations and give applications of these results to discrete Schro dinger operators on ℤN.

The 3G-inequality for general Schr�dinger operators on Lipschitz domains

In this paper we prove a new 3G-inequality for the Laplacian Green function on a bounded Lipschitz domain in ℝn, n≥3. We exploit this inequality to prove the existence and comparison of perturbed continuous Green functions associated with Δ−μ where μ is in a general class of signed Radon measures covering the well known Kato class.

A Schr�dinger Operator with Point Interactions on Sobolev Spaces

Schrodinger operators on Sobolev spaces are considered as new solvable models with point interactions. A simple formula for the deficiency indices of a minimal Schrodinger operator with point interactions is given. Examples of point interactions on the space W21(\({\cal R}\)3) are constructed.

A posteriori error estimator for the eigenvalue problem associated to the Schr�dinger operator with magnetic field

The ground state for the Neumann realization of the Schrodinger operator for constant and sufficiently large magnetic field presents a localization in the boundary of the domain and particularly in the corners where the angle is minimum. As the solution decreases exponentially fast away of the corner, it is rather difficult to catch it numerically. A natural idea is to try using a mesh refinement method coupled to a posteriori error estimates. The purpose of this paper is to provide such an estimator adapted to the problem.

Semi-classical resolvent estimates for the Schr�dinger operator on non-compact complete Riemannian manifolds

We prove uniform semi-classical estimates for the resolvent of the Schrodinger operator h 2Δ g + V (x), 0 0, where V is a real-valued non-negative potential and Δ g denotes the positive Laplace-Beltrami operator on a non-compact complete Riemannian manifold which may have a nonempty compact smooth boundary.

On the Absolutely Continuous Spectrum of Multi-Dimensional Schr�dinger Operators with Slowly Decaying Potentials

We consider a multi-dimensional Schrodinger operator −Δ+V in L2(R d ) and find conditions on the potential V which guarantee that the absolutely continuous spectrum of this operator is essentially supported by the positive real line. We prove some results which go beyond the case L1+L p with p<d.

Absolutely Continuous Spectrum of Schr�dinger Operators with Slowly Decaying and Oscillating Potentials

The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrodinger operator is essentially supported by [0, infi ...