ON THE ESSENTIAL SELF-ADJOINTNESS OF THE RELATIVISTIC HAMILTONIAN WITH A NEGATIVE SCALAR POTENTIAL

Source abstract

The relativistic quantum Hamiltonian H describing a spinless particle in an electromagnetic field is considered. H is associated with the classical Hamiltonian [Formula: see text] via Weyl’s correspondence. In the previous papers the second author has proved that H is essentially self-adjoint on [Formula: see text] if the scalar potential V(x) is a function bounded from below by a polynomial in x. In the present paper this result will be extended to show that H is essentially self-adjoint there if V(x) is bounded from below by -C exp a|x| for some positive constants C and a. Ameliorated is also the condition on the vector potential A(x). The result of this kind is quite different from that on the non-relativistic operator, i.e. the Schrödinger operator, but much closer to that on the Dirac operator.