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**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).

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