Embedded eigenvalues and resonances of Schrödinger operators with two channels

Wang, Xue Ping

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 16 (2007) no. 1, pp. 179-214.

Dans cet article, nous donnons dans le régime de perturbation une condition nécessaire et suffisante sur l’existence de valeurs propres plongées entre les deux seuils. Pour une valeur propre de l’opérateur non-perturbé plongée à un seuil, nous démontrons qu’elle peut engendrer à la fois des valeurs propres discrètes et des résonances.

In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.

[1] Agmon (S.).— Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), no. 2, p. 151-218 (1975). | Numdam | MR | Zbl

[2] Agmon (S.).— On perturbation of embedded eigenvalues. p. 1-14, in Progr. Nonlinear Differential Equations Appl., 21, Birkhäuser Boston, Boston, MA (1996). | MR | Zbl

[3] Agmon (S.), Herbst (I.), Skibsted (E.).— Perturbation of embedded eigenvalues in the generalized $N$-body problem. Comm. Math. Phys. 122, no. 3, p. 411-438 (1989). | MR | Zbl

[4] Aguilar (J.), Combes (J.M.).— A class of analytic perturbations for one-body Schrödinger Hamiltonians, Commun. in Math. Phys., 22, p. 269-279 (1971). | MR | Zbl

[5] Arai (A.).— Perturbation of embedded eigenvalues: a general class of exactly soluble models in Fock spaces. Hokkaido Math. J. 19, no. 1, p. 1-34 (1990). | MR | Zbl

[6] Derezinski (J.), Jaksic (V.).— Spectral theory of Pauli-Fierz operators, J. Funct. Analysis, 180, p. 243-327 (2001). | MR | Zbl

[7] Froese (R.), Herbst (I.).— Exponential bounds and absence of positive eigenvalues for $N$-body Schrödinger operators. Comm. Math. Phys. 87, no. 3, p. 429-447 (1982/83). | MR | Zbl

[8] Helffer (B.), Sjöstrand (J.).— Résonances en limite semi-classique, Bull. Soc Math. France, Mémoire No.24/25, p. 114 (1986). | Numdam | MR | Zbl

[9] Hörmander (L.).— The Analysis of Linear Partial Differential Operators II, Springer (1983). | MR | Zbl

[10] Howland (J.S.).— Puiseux series for resonances at an embedded eigenvalue. Pacific J. Math. 55, p. 157-176 (1974). | MR | Zbl

[11] Howland (J.S.).— The Livsic matrix in perturbation theory. J. Math. Anal. Appl. 50, p. 415-437 (1975). | MR | Zbl

[12] Isozaki (H.), Kitada (H.).— Microlocal resolvent estimates for $2$-body Schrödinger operators. J. Funct. Anal. 57, no. 3, p. 270-300 (1984). | MR | Zbl

[13] Jensen (A.), Kato (T.).— Spectral properties of Schrödinger operators and time decay of wave functions, Duke Math. J., 46, p. 583-611 (1979). | MR | Zbl

[14] Jensen (A.), Melgård (M.).— Perturbation of eigenvalues embedded at a threshold. Proc. Roy. Soc. Edinburgh Sect. A 132, no. 1, p. 163–179 (2002). | MR | Zbl

[15] Landau (L.), Lifchitz (E.).— Mécanique Quantique, Théorie Non Relativiste, Deuxième édition, Editions Mir, Moscou (1967). | Zbl

[16] Reed (M.), Simon (B.).— Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press (1978). | MR | Zbl

[17] Simon (B.).— Resonances in $n$-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math., p. 247-274 (1973). | MR | Zbl

[18] Sjöstrand (J.), Zworski (M.).— Complex scaling and the distribution of scattering poles, J. of Amer. Math. Soc., 4(4), p. 729-769 (1991). | MR | Zbl

[19] Sjöstrand (J.), Zworski (M.).— Elementary linear algebra for advanced spectral problems, preprint (2003).

[20] Soffer (A.), Weinstein (M.I.).— Time dependent resonance theory and perturbations of embedded eigenvalues. Partial differential equations and their applications (Toronto, 1995), p. 277–282, CRM Proc. Lecture Notes, 12, Amer. Math. Soc., Providence, RI (1997). | MR

[21] Wang (X.P.).— Microlocal resolvent estimates for N-body Schrödinger operators, J. of the Faculty of Sc., Univ. Tokyo, Sec. IA, 40, p. 337-385 (1993). | MR | Zbl

[22] Wang (X.P.).— Asymptotic behavior of the resolvent of N-body Schrödinger operators near a threshold, Ann. Henri Poincaré, 4, p. 553-600 (2003). | MR | Zbl

[23] Wang (X.P.).— Threshold energy resonance in geometric scattering, Matemática Contemporânea, 26, p. 135-164 (2004). | MR | Zbl

[24] Wang (X.P.), Wang (Y.F.).— Existence of two-cluster threshold resonances and the N-body Efimov effect, J. Math. Phys., 46(11) p. 112106-1 – 112106-12, (2005). | MR | Zbl