arXiv subjects

Shu Nakamura

Publications and source records attributed to Shu Nakamura.

40 records · Page 3Linked to original sources

arXiv2007

Singularities of solutions to Schrodinger equation on scattering manifold

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 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.

math.AP↗

arXiv2007

Semiclassical Singularity Propagation Property for Schrödinger Equations

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

math.AP↗

arXiv2007

Analytic wave front set for solutions to Schroedinger equation

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

math.AP↗

arXiv2004

Bounds on the spectral shift function and the density of states

We study spectra of Schrödinger operators on $\RR^d$. 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 $μ_n$ of the difference of the semigroups as $n\to \infty$ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger 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 Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.

math-ph↗