## 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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### References:

  Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2, 4, 151-218 (1975) · Zbl 0315.47007  Balslev, E., Analytic scattering theory for 2-body Schrödinger operator, J. Funct. Anal., 29, No. 3, 375-396 (1978) · Zbl 0392.47003  Balslev, E., The singular spectrum of elliptic operators in $$L^p(R^{n$$ · Zbl 0171.08501  Balslev, E.; Helffer, B., Limiting absorption principle and resonances for the Dirac operator, (Aarhus Universiteit Preprint (January 1990)) · Zbl 0756.35062  Balslev, E.; Skibsted, E., Resonance theory of two-body Schrödinger operators, Ann. Inst. H. Poincaré, 51, No. 2, 129-154 (1989) · Zbl 0714.35063  Baumogärtel, H., Endlich-dimensionale analytische Störungsteorie (1972), Akademie-Verlag: Akademie-Verlag Berlin  Berthier, A.; Georgescu, V., On the point spectrum of Dirac operators, J. Funct. Anal., 71, 309-338 (1987) · Zbl 0655.47043  Bjorken, J. D.; Drell, S. D., Relativistic Quantum Mechanics (1964), McGraw-Hill: McGraw-Hill New York · Zbl 0184.54201  Enss, V.; Thaller, B., Asymptotic observables and Coulomb scattering for the Dirac operator, Ann. Inst. H. Poincaré Phys. Théor., 45 (1986) · Zbl 0615.47008  Grigoire, D. R.; Nenciu, G.; Purice, R., On the nonrelativistic limit of the Dirac hamiltonian, Ann. Inst. H. Poincaré Phys. Théor., 51, No. 3, 231-263 (1989) · Zbl 0705.35115  Hachem, G., Théorie spectrale de l’opérateur de Dirac dans un champ électromangnétique à croissance linéaire à l’infini, Thèse d’état à l’université Paris-Nord (1988)  Helffer, B.; Sjöstrand, J., Résonances en limite semi-classique, Mémoire No. 24/25 de la SMF, Supplément du bulletin de la SMF, 114 (1986), Fasc. 3 · Zbl 0631.35075  Hunziker, W., On the nonrelativistic limit of the Dirac theory, Comm. Math. Phys., 40, 215-222 (1975)  Kato, T., Perturbation Theory for Linear Operators, (Grundlehren der mathematischen Wissenchraften, vol. 132 (1976), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0836.47009  Kuroda, S., An Introduction to Scattering Theory, (Lecture Notes, Vol. 51 (1980), Matematisk Institut: Matematisk Institut Aarhus) · Zbl 0407.47003  Osgood, W. F., (Lehrbuch der Functionentheorie, Vol. II (1965), Chelsea: Chelsea New York) · Zbl 0184.29701  Parisse, B., preprint de l’école Normale supérieure (1990) · Zbl 0723.35051  Seba, P., The complex scaling method for Dirac resonances, preprint Bibos, Lett. Math. Phys. (1987), to appear  Veselic, K., Perturbation of pseudoresolvents and analyticity in $$1c$$ in relativistic quantum mechanics, Comm. Math. Phys., 22, 27-43 (1971) · Zbl 0212.15701  Weder, R. A., Spectral properties of the Dirac Hamiltonian, Ann. Soc. Sci. Bruxelles Sér. 1, 87, 341-355 (1973)  Yamada, K., Eigenfunction expansions and scattering theory for Dirac operators, Publ. Res. Inst. Math. Sci., 11, 651-689 (1976) · Zbl 0334.35060
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