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Limiting absorption principle and resonances for the Dirac operator. (English) Zbl 0756.35062

The authors develop the stationary scattering theory based on the limiting absorbtion principle for the Dirac operator \(H_ D=-i\sum_ j^ 3\alpha_ j\partial/\partial x_ j+\beta+V(x)\), in \(L^ 2(\mathbb{R}^ 3,\mathbb{C}^ 4)\), where \(\alpha_ j\) and \(\beta\) are the Dirac matrices. Let \[ H_ 4^{t,s}=\{f:\mathbb{R}^ 3\to\mathbb{C}^ 4,\;\| f\|_{t,s}=\|(1+| x|^ 2)^{s/2}f\|_{H^ t_ 4}\}, \] where \(H^ t_ 4\) is an analogue of the Sobolev space.
Assume that \(\text{(A}_ 1)\) \(\exists\;s>1/2\) such that \(V\) is a compact operator from \(H_ 4^{1,-s}\) into \(H_ 4^{0,s}\),
\(\text{(A}_ 2)\) \((Vf,f)\in\mathbb{R}\) for \(f\in H_ 4^{1,-s}\), \(\text{(A}_ 3)\) \(V\in B(H_ 4^{1,t},H_ 4^{0,1+2s})\).
If \(V\) is a multiplication by \(v(x)\) then \(\sup_{x\in\mathbb{R}^ 3}\int_{| x-y|\leq 1}| v(y)|^ 2| x-y|^{-1- \varepsilon}dy<\infty\),
\(\int_{| x-y|\leq 1}| v(y)|^ 2\leq C| x|^{-1- \varepsilon}\), \(| x|>R\) for some \(\varepsilon,C,R\).
Theorem. The following limits exist for \(\lambda\in[(-\infty,- 1\cup[1,\infty)\backslash\sigma_ p(H_ D)\) in the uniform operator topology of \(B(H_ 4^{0,s},H_ 4^{1,-s})\), uniformly on compact sets: \[ R_ D(\lambda\pm iO)=\lim_{\varepsilon\downarrow 0}R_ D(\lambda\pm i\varepsilon)=R_{0D}(\lambda\pm iO)(1+VR_{0D}(\lambda\pm iO))^{-1} \] where \(R_{OD}\) is the resolvent of \(H_ D\mid_{V=0}\).
This is the limiting absorbtion principle in the sense of Agmon. The authors construct also the scattering matrix. After that, restricting themselves to potentials \(v(x)\), that can be splitted into a dilation analytic part and an exponentially decaying part they prove that the scattering matrix and the resolvent have the analytic continuation and define resonances as the common poles of resolvent and \(S\)-matrix and of the associated resonance functions. They prove an isomorphism between the space of resonance functions and the null space of the inverse \(S\)-matrix and establish that the dimension of these spaces is even just as the dimension of the eigenspaces.

MSC:

35P25 Scattering theory for PDEs
81U05 \(2\)-body potential quantum scattering theory
47A40 Scattering theory of linear operators
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