On the essential spectrum of the relativistic magnetic Schrödinger operator

Abstract

where λ ( ξ) = (|ξ − ( )|2 + 1)1/2. For simplicity we suppose here that the mass of the particle is equal to 1. All the differential and pseudodifferential operators considered in this paper are, possibly unbounded, operators in 2(R ), defined on the Schwartz space S of rapidly decreasing smooth functions on R . In [14] it was proved that if the derivatives of any positive order of ∈ ∞(R ; R ) are bounded, then (λ ) is essentially selfadjoint on S. Let be its unique selfadjoint extension. In [16] the authors proved that if itself is bounded and if all its derivatives converge to zero at infinity, then the essential spectrum of is equal to the essential spectrum of √ + 1, where is the quantum nonrelativistic magnetic Hamiltonian with vector potential , i.e. the selfadjoint operator generated by the differential operator ( − ( ))2. We shall prove in this paper that the essential spectra of and of √ + 1 are still equal if we drop the condition of boundedness of . Thus, vector potentials which behave at infinity as | |1−e, e positive and arbitrary small, are allowed. More precisely, the main result of the paper is the following theorem.