Tomio Umeda

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.

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.

THE DIRAC-HARDY AND DIRAC-SOBOLEV INEQUALITIES IN L 1

Dirac-Sobolev and Dirac-Hardy inequalities in

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

L^1

Asymptotic Behavior of the Resolvent of the Dirac Operator

are established in which the

Schnol’s Theorem and Spectral Properties of Massless Dirac Operators with Scalar Potentials

L^p

Representation formulas of the solutions to the Cauchy problems for first order systems

spaces which feature in the classical Sobolev and Hardy inequalities are replaced by weak

The zero modes and zero resonances of massless Dirac operators

L^p

The Asymptotic Limits of Zero Modes of Massless Dirac Operators

spaces. Counter examples to the analogues of the classical inequalities are shown to be provided by zero modes for appropriate Pauli operators constructed by Loss and Yau.

Scattering and spectral theory for the linear Boltzmann operator

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