## Relativistic Hamiltonians with dilation analytic potentials diverging at infinity.(English)Zbl 1252.81060

Summary: We investigate the spectral properties of the Dirac operator with a potential $$V(x)$$ and two relativistic Schrödinger operators with $$V(x)$$ and $$-V(x)$$, respectively. The potential $$V(x)$$ is assumed to be dilation analytic and to diverge at infinity. Our approach is based on an abstract theorem related to dilation analytic methods, and our results on the Dirac operator are obtained by analyzing dilated relativistic Schrödinger operators. Moreover, we explain some relationships of spectra and resonances between Schrödinger operators and the Dirac operator as the nonrelativistic limit.

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35P20 Asymptotic distributions of eigenvalues in context of PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35B34 Resonance in context of PDEs 35J10 Schrödinger operator, Schrödinger equation 47A20 Dilations, extensions, compressions of linear operators
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