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.


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


[1] 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
[3] Balslev, E., Analytic scattering theory for 2-body Schrödinger operator, J. Funct. Anal., 29, No. 3, 375-396 (1978) · Zbl 0392.47003
[4] Balslev, E., The singular spectrum of elliptic operators in \(L^p(R^{n\) · Zbl 0171.08501
[5] Balslev, E.; Helffer, B., Limiting absorption principle and resonances for the Dirac operator, (Aarhus Universiteit Preprint (January 1990)) · Zbl 0756.35062
[6] Balslev, E.; Skibsted, E., Resonance theory of two-body Schrödinger operators, Ann. Inst. H. Poincaré, 51, No. 2, 129-154 (1989) · Zbl 0714.35063
[7] Baumogärtel, H., Endlich-dimensionale analytische Störungsteorie (1972), Akademie-Verlag: Akademie-Verlag Berlin
[8] Berthier, A.; Georgescu, V., On the point spectrum of Dirac operators, J. Funct. Anal., 71, 309-338 (1987) · Zbl 0655.47043
[9] Bjorken, J. D.; Drell, S. D., Relativistic Quantum Mechanics (1964), McGraw-Hill: McGraw-Hill New York · Zbl 0184.54201
[10] Enss, V.; Thaller, B., Asymptotic observables and Coulomb scattering for the Dirac operator, Ann. Inst. H. Poincaré Phys. Théor., 45 (1986) · Zbl 0615.47008
[11] 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
[12] 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)
[13] 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
[14] Hunziker, W., On the nonrelativistic limit of the Dirac theory, Comm. Math. Phys., 40, 215-222 (1975)
[16] Kato, T., Perturbation Theory for Linear Operators, (Grundlehren der mathematischen Wissenchraften, vol. 132 (1976), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0836.47009
[17] Kuroda, S., An Introduction to Scattering Theory, (Lecture Notes, Vol. 51 (1980), Matematisk Institut: Matematisk Institut Aarhus) · Zbl 0407.47003
[18] Osgood, W. F., (Lehrbuch der Functionentheorie, Vol. II (1965), Chelsea: Chelsea New York) · Zbl 0184.29701
[19] Parisse, B., preprint de l’école Normale supérieure (1990) · Zbl 0723.35051
[20] Seba, P., The complex scaling method for Dirac resonances, preprint Bibos, Lett. Math. Phys. (1987), to appear
[21] Veselic, K., Perturbation of pseudoresolvents and analyticity in \(1c\) in relativistic quantum mechanics, Comm. Math. Phys., 22, 27-43 (1971) · Zbl 0212.15701
[22] Weder, R. A., Spectral properties of the Dirac Hamiltonian, Ann. Soc. Sci. Bruxelles Sér. 1, 87, 341-355 (1973)
[23] 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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