Searcharxiv⌕ Search

arXiv subjects

Shu Nakamura

Publications and source records attributed to Shu Nakamura.

At least 37 records · Page 2Linked to original sources
arXiv2017

A spectral shift function for Schrödinger operators with singular interactions

For the pair $\{-Δ, -Δ-αδ_\mathcal{C}\}$ of self-adjoint Schrödinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here $δ_{\cal{C}}$ denotes a singular $δ$-potential which is supported on a smooth compact hypersurface $\mathcal{C}\subset\mathbb{R}^n$ and $α$ is a real-valued function on $\mathcal{C}$.

math.SP↗
arXiv2016

Spectral shift functions and Dirichlet-to-Neumann maps

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, Schrödinger operators with compactly supported potentials, and finally, Schrödinger operators with singular potentials supported on hypersurfaces. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.

math.SP↗
arXiv2016

Microlocal resolvent estimates, revisited

Let $H$ be a Schrödinger type operator with long-range perturbation. We study the wave front set of the distribution kernel of $(H-λ\mp i0)^{-1}$, where $λ$ is in the absolutely continous spectrumof $H$.The result is a refinement of the microlocal resolvent estimate of Isozaki-Kitada \cite{IK1,IK2}. We prove the result for a class of pseudodifferential operators on manifolds so that they apply to discrete Schrödinger operators and higher order operators on the Euclidean space. The proof relies on propagation estimates, whereas the original proof of Isozaki-Kitada relies on a construction of parametrices.

math.AP↗
arXiv2014

Microlocal properties of scattering matrices

We consider scattering theory for a pair of operators $H_0$ and $H=H_0+V$ on $L^2(M,m)$, where $M$ is a Riemannian manifold, $H_0$ is a multiplication operator on $M$ and $V$ is a pseudodifferential operator of order $-μ$, $μ>1$. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, but it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.

math.AP↗
arXiv2013

A Remark on the Mourre Theory for Two Body Schrödinger operators

On this short note, we apply the Mourre theory of the limiting absorption with {\it difference} type conditions on the potential, instead of conditions on the derivatives. In order that we modify the definition of the conjugate operator, and we apply the standard abstract Mourre theory. We also discuss examples to which the method applies.

math-ph↗
arXiv2013

Propagation of singularities for Schrödinger equations with modestly long range type potentials

In a previous paper by the second author, we discussed a characterization of the microlocal singularities for solutions to Schrödinger equations with long range type perturbations, using solutions to a Hamilton-Jacobi equation. In this paper we show that we may use Dollard type approximate solutions to the Hamilton-Jacobi equation if the perturbation satisfies somewhat stronger conditions. As applications, we describe the propagation of microlocal singularities for $e^{itH_0}e^{-itH}$ when the potential is asymptotically homogeneous as $|x|\to\infty$, where $H$ is our Schrödinger operator, and $H_0$ is the free Schrödinger operator, i.e., $H_0=-\frac12 \triangle$. We show $e^{itH_0}e^{-itH}$ shifts the wave front set if the potential $V$ is asymptotically homogeneous of order 1, whereas $e^{itH}e^{-itH_0}$ is smoothing if $V$ is asymptotically homogenous of order $β\in (1,3/2)$.

math.AP↗
arXiv2012

The Spectral Shift Function and The Friedel Sum Rule

We study the relationship between the spectral shift function and the excess charge in potential scattering theory. Although these quantities are closely related to each other, they have been often formulated in different settings so far. Here we first give an alternative construction of the spectral shift function, and then we prove that the spectral shift function thus constructed yields the Friedel sum rule.

math-ph↗
arXiv2012

Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds

Let $M$ be a scattering manifold, i.e., a Riemannian manifold with asymptotically conic structure, and let $H$ be a Schrödinger operator on $M$. We can construct a natural time-dependent scattering theory for $H$ with a suitable reference system, and the scattering matrix is defined accordingly. We here show the scattering matrices are Fourier integral operators associated to a canonical transform on the boundary manifold generated by the geodesic flow. In particular, we learn that the wave front sets are mapped according to the canonical transform. These results are generalizations of a theorem by Melrose and Zworski, but the framework and the proof are quite different. These results may be considered as generalizations or refinements of the classical off-diagonal smoothness of the scattering matrix for 2-body quantum scattering on Euclidean spaces.

math.AP↗
arXiv2012

The spectrum of the scattering matrix near resonant energies in the semiclassical limit

The object of study in this paper is the on-shell scattering matrix $S(E)$ of the Schrödinger 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 $E_\text{res}-\varepsilon$ to $E_\text{res}+\varepsilon$, where $E_\text{res}$ is a real part of a resonance, and $\varepsilon$ 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.

math.SP↗
arXiv2011

Understanding the Random Displacement Model: From Ground-State Properties to Localization

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.

math-ph↗
arXiv2011

Localization for the random displacement model

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".

math-ph↗
arXiv2009

Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

We consider Schrödinger operators $H$ on $R^n$ with variable coefficients. Let $H_0=-\frac12\triangle$ be the free Schrödinger operator and we suppose $H$ is a "short-range" perturbation of $H_0$. Then, under the nontrapping condition, we show the time evolution operator: $e^{-itH}$ can be written as a product of the free evolution operator $e^{-itH_0}$ 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.

math.AP↗
arXiv2009

Lifshitz Tails for Generalized Alloy Type Random Schrödinger Operators

We study Lifshitz tails for random Schrödinger operators where the random potential is alloy type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose the single site potentials are distributed in a finite set of functions, and we show that under suitable symmetry conditions, they have Lifshitz tail at the bottom of the spectrum except for special cases. When the single site potential is symmetric with respect to all the axes, we give a necessary and sufficient condition for the existence of Lifshitz tails. As an application, we show that certain random displacement models have Lifshitz singularity at the bottom of the spectrum, and also complete the study of continuous Anderson type models undertaken in arXiv : 0804.4079

math-ph↗
arXiv2008

Wave front set for solutions to perturbed harmonic oscillators

We consider Schrödinger equations with variable coefficients and the harmonic potential. We suppose the perturbation is short-range type in the sense of [Nakamura 2004]. 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 $t=π$.

math.AP↗
arXiv2008

Time-dependent scattering theory for Schrödinger operators on scattering manifolds

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.

math-ph↗
arXiv2008

Spectral extrema and Lifshitz tails for non monotonous alloy type models

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.

math-ph↗
arXiv2008

Analytic Wave Front Set for Solutions to Schrödinger Equations II -- Long Range Perturbations

This paper is a continuation of a paper by the authors: arXiv:0706.0415, where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results of the paper to long-range perturbations (in particular, we can allow potentials growing like $ ^{2-\varepsilon}$ at infinity). More precisely, we construct a modified quantum free evolution $G_0(-s, hD_z)$ acting on Sjöstrand's spaces, and we characterize the analytic wave front set of the solution $e^{-itH}u_0$ of the Schrödinger equation, in terms of the semiclassical exponential decay of $G_0(-th^{-1}, hD_z)T u_0$, 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. It is an extension of a paper by Nakamura (arXiv:math/0605742) to the analytic framework.

math.AP↗