Shu Nakamura

Let $X=\mathbb{R}\times M$ be the spacetime, where $M$ is a closed manifold equipped with a Riemannian metric $g$, and we consider a symmetric Klein-Gordon type operator $P$ on $X$, which is asymptotically converges to $\partial_t^2-\triangle_g$ as $|t|\to\infty$, where $\triangle_g$ is the Laplace-Beltrami operator on $M$. We prove the essential self-adjointness of $P$ on $C_0^\infty(X)$. The idea of the proof is closely related to a recent paper by the authors on the essential self-adjointness for Klein-Gordon operators on asymptotically flat spaces.

Here we discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type, previously proved by Vasy (2020) and Nakamura-Taira (2021). For simplicity, here we discuss the second order cases, i.e., Klein-Gordon type operators only.

We consider the quantum graph Hamiltonian on the square lattice in Euclidean space, and we show that the spectrum of the Hamiltonian converges to the corresponding Schrödinger operator on the Euclidean space in the continuum limit, and that the corresponding eigenfunctions and eigenprojections also converge in some sense. We employ the discrete Schrödinger operator as the intermediate operator, and we use a recent result by the second and third author on the continuum limit of the discrete Schrödinger operator.

We propose a method of data quantization of finite discrete-time signals which optimizes the error estimate of low frequency Haar coefficients. We also discuss the error/noise bounds of this quantization in the Fourier space. Our result shows one can quantize any discrete-time analog signal with high precision at low frequencies. Our method is deterministic, and it employs no statistical arguments, nor any probabilistic assumptions.

We consider scattering matrix for Schr\&quot;odinger-type operators on $R^d$ with<br /> perturbation $V(x)=O(\langle x\rangle^{-1})$ as $|x|\to\infty$. We show that<br /> the scattering matrix (with time-independent modifiers) is a pseudodifferential<br /> operator. We present examples of which the spectrum of the scattering matrix is<br /> dense point spectrum.

We show that the scattering matrix for a class of Schr\&quot;odinger-type<br /> operators with long-range perturbations is a Fourier integral operator with the<br /> phase function which is the generating function of the modified classical<br /> scattering map.

For the pair ˆ’Δ, ˆ’Δ ˆ’αδC of self-adjoint Schröodinger operators in L2(ℝn) a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here δC denotes a singular δ-potential which is supported on asmooth compact hypersurface C ⊂ (ℝn) and δ is a real-valued function on C.

The spectral shift function of a pair of self-adjoint operators is expressed<br /> via an abstract operator valued Titchmarsh--Weyl $m$-function. This general<br /> result is applied to different self-adjoint realizations of second-order<br /> elliptic partial differential operators on smooth domains with compact<br /> boundaries, Schr\&quot;{o}dinger operators with compactly supported potentials, and<br /> finally, Schr\&quot;{o}dinger operators with singular potentials supported on<br /> hypersurfaces. In these applications the spectral shift function is determined<br /> in an explicit form with the help of (energy parameter dependent)<br /> Dirichlet-to-Neumann maps.

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

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

We consider the scattering theory for discrete Schr\&quot;odinger operators on<br /> $Z^d$ with long-range potentials. We prove the existence of modified wave<br /> operators constructed in terms of solutions of a Hamilton-Jacobi equation on<br /> the torus $T^d$.

In a previous paper by the second author [11], we discussed a characterization of the microlocal singularities for solutions to Schrodinger 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(itH0)e(-itH) when the potential is asymptotically homogeneous as vertical bar x vertical bar -&gt; infinity, where H is our Schrodinger operator, and H-0 is the free Schrodinger operator, i.e., H-0 = -1/2 Delta. We show e(itH0)e(-itH) shifts the wave front set if the potential V is asymptotically homogeneous of order 1, whereas e(itH)e(-itH0) is smoothing if V is asymptotically homogeneous of order beta epsilon (1, 3/2).

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.

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

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

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

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

This is a summary of recent work, jointly with J. Baker, F. Klopp and S. Nakamura, concerning the random displacement model resulting in a proof of localization near the edge of the deterministic spectrum. Localization is meant in both senses, pure point spectrum with exponentially decaying eigenfunctions as well as dynamical localization. The proof relies on a well established multiscale analysis and the main problem is to verify the necessary ingredients, such as a Lifshitz tail estimate and a Wegner estimate.

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 &lt; x &gt; 2-epsilon at infinity). More precisely, we construct a modified quantum free evolution G0(-s, hDz) acting on Sjostrand&apos;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.

We study Lifshitz tails for random Schrodinger 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 a 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 a Lifshitz singularity at the bottom of the spectrum, and also complete our previous study (2009) of continuous Anderson type models.

We consider Schr\&quot;odinger operators $H$ on $R^n$ with variable coefficients.<br /> Let $H_0=-\frac12\triangle$ be the free Schr\&quot;odinger operator and we suppose<br /> $H$ is a &quot;short-range&quot; perturbation of $H_0$. Then, under the nontrapping<br /> condition, we show the time evolution operator: $e^{-itH}$ can be written as a<br /> product of the free evolution operator $e^{-itH_0}$ and a Fourier integral<br /> operator $W(t)$, which is associated to the canonical relation given by the<br /> classical mechanical scattering. We also prove a similar result for the wave<br /> operators. These results are analogous to results by Hassell and Wunsch, but<br /> the assumptions, the proof and the formulation of results are considerably<br /> different. The proof employs an Egorov-type theorem similar to those used in<br /> previous works by the authors combined with a Beals-type characterization of<br /> Fourier integral operators.

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

This paper is a continuation of a previous paper by the same authors, where<br /> an analytic smoothing effect was proved for long-range type perturbations of<br /> the Laplacian $H_0$ on $\re^n$. In this paper, we consider short-range type<br /> perturbations $H$ of the Laplacian on $\re^n$, and we characterize the analytic<br /> wave front set of the solution to the Schr\&quot;odinger equation:<br /> $e^{-itH}f$, in terms of that of the free solution: $e^{-itH_0}f$, for $t&lt;0$<br /> in the forward nontrapping region. The same result holds for $t&gt;0$ in the<br /> backward nontrapping region. This result is an analytic analogue of results by<br /> Hassel and Wunsch and Nakamura.

We consider Schr\&quot;odinger equations with variable coefficients, and it is<br /> supposed to be a long-range type perturbation of the flat Laplacian on $R^n$.<br /> We characterize the wave front set of solutions to Schr\&quot;odinger equations in<br /> terms of the initial state. Then it is shown that the singularity propagates<br /> following the classical flow, and it is formulated in a semiclassical settings.<br /> Methods analogous to the long-range scattering theory, in particular a modified<br /> free propagator, are employed.

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 --&gt; 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.