Liens entre les résonances pour l’opérateur de Dirac et de Schrödinger. (Links between resonances of Schrödinger and Dirac operators). (French) Zbl 0713.35071

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1990, No. 12, 7 p. (1990).
The Schrödinger operator \(P(h)=-(1/2m)h^ 2\Delta +V(x)\) describes a non relativistic particle of mass m and the Dirac operator \(D(h)=hc\sum^{3}_{j=1}\alpha_ jD_ j+mc^ 2\alpha_ 4+VI_ 4\) describes particles of spin 1/2 in restricted relativity, which, in the classical case, tends to the Newton’s mechanics as the light velocity c tends to \(+\infty\) (here the \(\alpha_ j\) are hermitics anticommuting \(4\times 4\) matrices). It is also natural to search links between resonances of Schrödinger and Dirac operators.
We use the Helffer-Sjöstrand’s theory to define resonances for h small for the Schrödinger operator and adapt this theory for the Dirac operator to obtain a “formal” link between resonances of these operators. The main difference is that the multiplicity of Dirac’s resonances is even (see also the work of Balsev-Helffer).
We are now interested in “shape” resonances because we can calculate the imaginary part of such resonances (recall that the imaginary part of a resonance describes the lifetime of a resonant state) by WKB methods. One can obtain a result in the case of a radial potential of “non degenerated well in an island” type: \[ \lim_{\substack{ c\to\infty\\ h\to 0}} Im z_ D/Im z_ S=1, \] where \(z_ D\) is the resonance for the Dirac operator corresponding to the first eigenvalue of the well, \(z_ S\) is the Schrödinger’s resonance, and equ means an equivalent (as h tends to 0).
Reviewer: B.Parisse


35P99 Spectral theory and eigenvalue problems for partial differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35Q40 PDEs in connection with quantum mechanics
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