Shu Nakamura

The norm resolvent convergence of discrete Schrödinger operators to a continuum Schrödinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum with respect to the Hausdorff distance.

The resonances for the Wigner-von Neumann type Hamiltonian are defined by the periodic complex distortion in the Fourier space. Also, following Zworski, we characterize resonances as the limit points of discrete eigenvalues of the Hamiltonian with a quadratic complex absorbing potential in the viscosity type limit.

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

The object of study in this paper is the on-shell scattering matrix S(E) of the Schrodinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of S(E) in the semiclassical limit when the energy parameter E varies from Eres - ε to Eres + ε, where Eres is a real part of a resonance and ε is sufficiently small. The main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle. This result is closely related to the Breit-Wigner effect. © 2013 American Mathematical Society.

We prove spectral and dynamical localization for the multi-dimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as "bubbles tend to the corners".

We give a detailed survey of results obtained in the most recent half-decade which led to a deeper understanding of the random displacement model, a model of a random Schrödinger operator which describes the quantum mechanics of an electron in a structurally disordered medium. These results started by identifying configurations which characterize minimal energy, then led to Lifshitz tail bounds on the integrated density of states as well as a Wegner estimate near the spectral minimum, which ultimately resulted in a proof of spectral and dynamical localization at low energy for the multi-dimensional random displacement model.

We consider Schrödinger operators H on ℝ n with variable coefficients. Let H o = -1/2δ be the free Schrödinger operator and we suppose H is a "short-range" perturbation of H o. Then, under the nontrapping condition, we show that the time evolution operator: e -itH can be written as a product of the free evolution operator e-itH o and a Fourier integral operator W(t) which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results are analogous to results by Hassell and Wunsch, but the assumptions, the proof and the formulation of results are considerably different. The proof employs an Egorov-type theorem similar to those used in previous works by the authors combined with a Beals-type characterization of Fourier integral operators.

We construct a time-dependent scattering theory for Schrödinger operators on a manifold $M$ with asymptotically conic structure. We use the two-space scattering theory formalism, and a reference operator on a space of the form $R\times \partial M$, where $\partial M$ is the boundary of $M$ at infinity. We prove the existence and the completeness of the wave operators, and show that our scattering matrix is equivalent to the absolute scattering matrix, which is defined in terms of the asymptotic expansion of generalized eigenfunctions. Our method is functional analytic, and we use no microlocal analysis in this paper.

This paper is a continuation of [9], where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of [9] to long-range perturbations (in particular, we can allow potentials growing like < x > 2-epsilon at infinity). More precisely, we construct a modified quantum free evolution G0(-s, hDz) acting on Sjostrand's spaces, and we characterize the analytic wave front set of the solution e-itHu0 of the Schrodinger equation, in terms of the semiclassical exponential decay of G0(-th-1, hDz)Tu0, where T stands for the Bargmann-transform. The result is valid for t0 near the forward non trapping points, and for t0 near the backward non trapping points. It is an extension of [12] to the analytic framework.

In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e.. noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in terms of the initial condition and the classical scattering maps under the nontrapping condition. Our result is closely related to a recent work by Hassell and Wunsch, though our model is more general and the method, which relies heavily on scattering theoretical ideas, is simple and quite different. In particular, we use an Egorov-type argument in the standard pseudodifferential symbol classes, and avoid using Legendre distributions. In the proof, we employ a microlocal smoothing property in terms of the radially homogenous wave front set, which is more precise than the preceding results.

This paper is a continuation of [A. Martinez, S. Nakamura, V. Sordoni, Analytic smoothing effect for the Schrodinger equation with long-range perturbation, Comm. Pure Appl. Math. LIX (2006) 1330-1351], where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian H(0) on R(n). In this paper, we consider short-range type perturbations H of the Laplacian on R(n), and we characterize the analytic wave front set of the solution to the Schrodinger equation: e(-itH) f, in terms of that of the free solution: e(-itH0) f, for t < 0 in the forward non-trapping region. The same result holds for t > 0 in the backward non-trapping region. This result is an analytic analogue of results by Hassel and Wunsch [A. Hassel, J. Wunsch, The Schrodinger propagator for scattering metrics, Ann. of Math. 162 (2005) 487-523] and Nakamura [S. Nakamura, Wave front set for solutions to Schrodinger equations, J. Funct. Anal. 256 (2009) 1299-1309]. (C) 2009 Elsevier Inc. All rights reserved.

In the present note, we determine the ground state energy and study the existence of Lifshitz tails near this energy for some non monotonous alloy type models. Here, non monotonous means that the single site potential coming into the alloy random potential changes sign. In particular, the random operator is not a monotonous function of the random variables.

We consider solutions to Schrodinger equation on R(d) with variable coefficients. Let H be the Schrodinger operator and let u(t) = e(-itH)u(0) be the solution to the Schrodinger equation with the initial condition u(0) is an element of L(2)(R(d)). We show that the wave front set of u(t) in the nontrapping region can be characterized by the wave front set of e(-itH0)u(0). where H(0) is the free Schrodinger operator. The characterization of the wave front set is given by the wave operator for the corresponding classical mechanical scattering (or equivalently, by the asymptotics of the geodesic flow). (C) 2008 Elsevier Inc. All rights reserved.

We consider Schrodinger equations with variable coefficients and the harmonic potential. We suppose the perturbation is short-range type in the sense of [7]. We characterize the wave front set of the solutions to the equation in terms of the classical scattering data and the propagator of the unperturbed harmonic oscillator. In particular, we give a recurrence of singularities type theorem for the propagation of the period t=.

We consider Schrodinger equations with variable coefficients, which are long-range type perturbations of the flat Laplacian on R-n. We characterize the wave front set of solutions to Schrodinger equations in terms of the initial state. Then it is shown that the singularities propagates along the classical flow, and results are formulated in a semiclassical setting. Methods analogous to the long-range scattering theory, in particular a modified free propagator, are employed.

We consider the Schrodinger equation associated to long range perturbations of the flat Euclidean metric (in particular, potentials growing subquadratically at infinity are allowed). We construct a modified quantum free evolution G(0)(s) acting on Sjostrand's spaces, and we characterize the analytic wave front set of the solution e(-itH) u(0) of the Schrodinger equation, in terms of the semiclassical exponential decay of G(0)(-th(-1))Tu(0), where T stands for the Bargmann-transform. The result is valid for t < 0 near the forward non-trapping points, and for t > 0 near the backward non-trapping points.

We study the microlocal analytic singularity of solutions to the Schrodinger equation with analytic coefficients. Using microlocal weight estimates developed for estimating phase space tunneling, we prove microlocal smoothing estimates that generalize results by Robbiano and Zuily. We show that the exponential decay of the initial state in a cone in the phase space implies microlocal analytic regularity of the solution at a positive time. We suppose the Schrodinger operator is a long-range-type perturbation of the Laplacian, and we employ positive commutator-type estimates to prove the smoothing property. (c) 2005 Wiley Periodicals, Inc.

We study spectra of Schrodinger operators on R-d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values mu(n) of the difference of the semigroups as n --> infinity and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrodinger operators. The single site potential u is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Holder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Holder continuity of the integrated density of states.

In this paper we study the propagation. of singularity for Schrodinger-type equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, which propagates along straight lines with finite speed away from x not equal 0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the microlocal smoothing property of Craig, Kappeler, and Strauss under more general assumptions.

We give a simple new proof of the exponential decay estimate in the adiabatic theory. The idea is to combine the stationary scattering theory for time-dependent Hamiltonian with tunneling estimates in the energy space.

We show that the Schrodinger propagator can be expanded in terms of resonances at energy levels at which a barrier separates the interaction region from infinity. The expansions hold for all times with errors small in the semi-classical parameter. As a byproduct we obtain a result on the approximation of clusters of resonant states by clusters of eigenfunctions of a self-adjoint reference operator. (C) 2002 Elsevier Inc. All rights reserved.

This short note is devoted to the proof of Lifshitz tails and a Wegner estimate, and thus, band edge localization, for the random hopping model. (C) 2003 American Institute of Physics.

We prove Anderson localization near the bottom of the spectrum for two-dimensional discrete Schrodinger operators with random magnetic fields and no scalar potentials. We suppose the magnetic fluxes vanish in pairs, and the magnetic field strength is bounded from below by a positive constant. Main lemmas are the Lifshitz tail and the Wegner estimate on the integrated density of states. Then, Anderson localization, i.e., pure point spectrum with exponentially decreasing eigenfunctions, is proved by the standard multiscale argument.

We prove exponentially small estimates on the off-diagonal terms of the scattering matrix associated to two-state semiclassical Schrodinger Hamiltonians. Our method is based on phase space tunneling estimates and a splitting of the operator by means of Toeplitz-type phase space cutoff operators. (C) 2002 Elsevier Science (USA).

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Holder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on the L-P-theory of the spectral shift function (SSF), for p greater than or equal to 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal I-P, for 0 < p less than or equal to 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local Holder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.

In this note, we consider the Lifshitz singularity for Schrodinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrodinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.

We develop the L-p-theory of the spectral shift function, for p greater than or equal to 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal I-p, for 0 < p less than or equal to 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Holder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the single-site potential, local Holder continuity is proved at all energies. This new Wegner estimate, together with other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations.

We prove an estimate on the difference of the number of eigenvalues for Schrodinger operators with Dirichlet and Neumann boundary conditions in large boxes. In the proof, we use Krein's theory of the spectral shift function. As an application, we show that the integrated density of states is independent of the choice of boundary conditions. (C) 2001 Academic Press.

We study the behavior of the density states at the lower edge of the spectrum for Schrodinger operators with random magnetic fields. We use a new estimate on magnetic Schrodinger operators, which is similar to the Avron-Herbst-Simon estimate but the bound is always nonnegative.

Lifshitz tail for 2 dimensional discrete Schrodinger operator with Anderson-type random magnetic field is proved. We first prove local energy estimates for deterministic discrete magnetic Schrodinger operators, and then follow the large deviation argument of Simon [6].

Semiclassical asymptotics of the spectral shift function (SSF) for Schrodinger operator is studied at trapping energies. It is shown that the SSF converges to sum of a smooth function and a step function, which is essentially the counting function of resonances. In particular, the Weyl asymptotics is proved.