Yoshimi Saitō

Convergence of the Neumann Laplacian on shrinking domains

holds for some specific domains (Example 4.8 and §5), where 7 is the trace operator from f2 = Hj, into a weighted space on the tree F, 2: e C\[0,00), and, for a function / on f2, is the restriction of / on Jl^.

An asymptotic behavior of the S‐matrix and the inverse scattering problem

An asymptotic formula of the S‐matrix as k→∞ will be shown for the general short‐range scattering in RN(N≥2). The main tool for the proof is the spectral decomposition theorem for the Schrodinger operator. By the use of the asymptotic formula, the uniqueness of the inverse scattering problem will be shown and a reconstruction formula for the potential will be presented for the general short‐range scattering.

An approximation formula in the inverse scattering problem

An approximate formula is given by which the potential Q(x) can be recovered with any accuracy if only the scattering data of the Schrodinger operator −Δ+Q(x) is known for some sufficiently high energy value k. Thus the scattering data for all k around k=∞ is not needed in order to get a good approximate value for the potential. The main tools for the proof are an asymptotic formula for the S‐matrix [Y. Saitō, J. Math. Phys. 25, 3105 (1984)] and the spectral decomposition theorem for the Schrodinger operator −Δ+Q(x) based on the limiting absorption principle.

EIGENFUNCTIONS AT THE THRESHOLD ENERGIES OF MAGNETIC DIRAC OPERATORS

Discussed are ±m modes and ±m resonances of Dirac operators with vector potentials HA = α · (D - A(x)) + mβ. Asymptotic limits of ±m modes at infinity are derived when |A(x)| ≤ C〈x〉-ρ, ρ > 1, provided that HA has ±m modes. In wider classes of vector potentials, sparseness of the vector potentials which give rise to the ±m modes of HA are established. It is proved that no HA has ±m resonances if |A(x)| ≤ C〈x〉-ρ, ρ > 3/2.

On the spectrum of the reduced wave operator with cylindrical discontinuity

Consider the differential operator H = -(1/m(x))L, where L is the N-dimensional Laplacian, in the weighted Hilbert space of square integrable functions on N-dimensional Euclidean space with weight m(x)dx. Here m(x) is a positive step function with a surface S of discontinuity (the separation surface). So far the stratified media in which the separating surface S consists of paralell planes have been vigorously studied. Also the case where S has a cone shape has been discussed. In this work we shall deal with a new type of discontinuity which we call cylindrical discontinuity. Under this condition we shall use the limiting absorption method to prove that H is absolute continuous. Our method is based on a apriori estimates of radiation condition term.

RESOLVENT ESTIMATES OF THE DIRAC OPERATOR

We shall investigate the asymptotic behavior of the extended resolvent R(s) of the Dirac operator as |s| increases to infinity, where s is a real parameter. It will be shown that the norm of R(s), as a bounded operator between two weighted Hilbert spaces of square integrable functions on the 3-dimensional Euclidean space, stays bounded. Also we shall show that R(s) converges 0 strongly as |s| increases to infinity. This result and a result of Yamada [15] are combined to indicate that the extended resolvent of the Dirac operator decays much more slowly than those of Schroedinger operators.

A Sequence of Zero Modes of Weyl–Dirac Operators and an Associated Sequence of Solvable Polynomials

It is shown that a series of solvable polynomials is attached to the series of zero modes constructed by Adam, Muratori and Nash [1].

Asymptotic Behavior of the Resolvent of the Dirac Operator

We consider the Dirac operator

The principle of limiting absorption for the non-selfadjoint Schrödinger operator in R2

H = - i\sum\limits_{j = 1}^3 {{\alpha _j}\frac{\partial }{{\partial {x_j}}} + \beta } + Q(x),