On the essential spectrum of the relativistic magnetic Schrödinger operator.
Introduction: In a series of papers one has studied the Weyl quantized Hamiltonian of a relativistic spinless particle with a magnetic vector potential : \[ [Op^w (\lambda_a)u](x) = (2\pi)^{-n}\int\int e^{i\langle x-y,\xi\rangle} \lambda_a\biggl(\frac{x+y}{2},\xi\biggr)u(y)\,dy\,d\xi, \] where \(\lambda_a(x,\xi) = (|\xi -a(x)|^2 + 1)^{1/2}.\) 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 \(L^2(\mathbb R^n)\), defined on the Schwartz space \(\mathcal S\) of rapidly decreasing smooth functions on \(\mathbb R^n\). In [\textit{M. Nagase} and \textit{T. Umeda}, J. Funct. Anal. 92, No. 1, 136--154 (1990; Zbl 0715.46051)] it was proved that if the derivatives of any positive order of \(a\in C^\infty(\mathbb R^n;\mathbb R^n)\) are bounded, then \(Op^w (\lambda_a)\) is essentially selfadjoint on \(\mathcal S\). Let \(h_a\) be its unique selfadjoint extension. In [Osaka J. Math. 30, No. 4, 839--853 (1993; Zbl 0814.35138)] \textit{M. Nagase} and \textit{T. Umeda} proved that if \(a\) itself is bounded and if all its derivatives converge to zero at infinity, then the essential spectrum of \(h_a\) is equal to the essential spectrum of \(\sqrt{H_a+1}\), where \(H_a\) is the quantum nonrelativistic magnetic Hamiltonian with vector potential, i.e. the selfadjoint operator generated by the differential operator \((D-a(X))^2\). We prove in this paper that the essential spectra of \(h_a\) and of \(\sqrt{H_a+1}\) are still equal if we drop the condition of boundedness of \(a\). Thus, vector potentials which behave at infinity as \(|x|^{1-\varepsilon}\), \(\varepsilon\) positive and arbitrary small, are allowed. More precisely, the main result of the paper is the following. Theorem 1.1. Suppose that: (i) the vector potential \(a\in C^\infty(\mathbb R^n;\mathbb R^n)\) is such that \[ \lim_{|x|\to\infty} \partial\beta a(x) = 0\qquad \forall \beta\in\mathbb N, \beta > 0; \] (ii) the scalar potential \(V\) is a continuous function such that \(\lim_{|x|\to\infty} V(x) = 0\). If \(h_a\) is the unique selfadjoint extension of \(Op^w (\lambda_a)\) and \(H_a\) is the unique selfadjoint extension of \((D-a(X))^2\), then \[ \sigma_{\text{ess}}(h_a+V) = \sigma_{\text{ess}}[(H_a+ 1)^{1/2}]. \] Let us also mention, that as a by-product of our proof, we not only recover some results from Nagase and Umeda (1998, 1992) concerning the essential selfadjointness of some pseudodifferential operators, but we can also obtain the domain of definition of the generated self-adjoint operator (Theorem 3.4). We now give the plan of the paper. In the second section we recall some results on the calculus of pseudodifferential operators. These results will be used in the next two sections. In the third one, we introduce what we call the \(a\)-magnetic Sobolev spaces, a particular case of weighted Sobolev spaces defined by Beals. As a corollary of the results proved in this section, we obtain that the domain \(h_a\) of is equal to the form domain of \(H_a\). If all the derivatives of \(a\) are bounded, it is also proved that the \(a\)-magnetic Sobolev space of order \(m\) is equal to the domain of \(H_a^{m/2}\). We think that this kind of results is already known, but we never saw them explicitly stated. The last section contains the proof of the main theorem.
scientific article; zbMATH DE number 1874594