J. Shabani and A. Vyabandi
abstract:
We introduce and study a new 2-parameter model of relativistic point
interactions in one dimension formally given by
\begin{eqnarray*}
D_{{\underline{\underline \alpha}},y} =D + {\underline{\underline \alpha}}\delta(x
- y);x\in \IR, \;\; y>0,
\end{eqnarray*}
where $D$ is the free Dirac Hamiltonian and ${\underline{\underline \alpha}}$ is
a $2 \times 2$ matrix. $D_{{\underline{\underline \alpha}},y}$ provides a
generalization of two models of relativistic point interactions discussed in [Lett.
Math. Phys. {\bf{13}} (1987), 345--358].
We define $D_{{\underline{\underline \alpha}},y}$ using the theory of self-adjoint
extensions of symmetric closed operators in Hilbert spaces, derive its resolvent
equation, analyze its spectral properties and discuss scattering theory for the
pair $(D_{{\underline{\underline \alpha}},y},D)$. We also study the
nonrelativistic limit of $D_{{\underline{\underline \alpha}},y}$ which provides
a special 2-parameter model of the one-dimensional generalized point
interactions introduced in [P. Exner and H. Grosse, Some properties of the one
dimensional
generalized point interactions. Preprint, 1999].
Mathematics Subject Classification: 81Q10, 47N50, 81Q05
Key words and phrases: boundary conditions problem, one-dimensional Dirac operator, self-adjoint extensions, resolvent equation, spectral properties, nonrelativistic limit, scattering theory