Asymptotic expansion in time of the Schrödinger group on conical manifolds

Wang, Xue Ping

Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1903-1945.

Nous étudions la contribution des états résonnants d’énergie nulle aux singularités de la résolvante près de zéro de l’opérateur de Schrödinger $P$ sur les variétés riemanniennes à bout conique. Sous une condition non-captive à haute énergie, nous obtenons le développement asymptotique du groupe de Schrödinger $U\left(t\right)={\mathrm{e}}^{-itP}$ pour $t$ grand.

For Schrödinger operator $P$ on Riemannian manifolds with conical end, we study the contribution of zero energy resonant states to the singularity of the resolvent of $P$ near zero. Long-time expansion of the Schrödinger group $U\left(t\right)={\mathrm{e}}^{-itP}$ is obtained under a non-trapping condition at high energies.

Classification : 35P25, 47A40, 81U10
Mots clés : Resolvent expansion, zero energy resonance, Schrödinger operator with metric

[1] Albeverio, S.; Bollé, D.; Gesztesy, F.; Hoegh-Krohn, R. Low-energy parameters in nonrelativistic scattering theory, Ann. Physics, Volume 148 (1983), pp. 308-326 | DOI | MR | Zbl

[2] Amado, R. D.; Greenwood, F. C. There is no Efimov effect for four or more particles, Phys. Rev. D, Volume 7 (1973), pp. 2517-2519 | DOI | MR

[3] Amado, R. D.; Noble, J. V. On Efimov’s effect: A new pathology of three-particle systems, Phys. Lett. B, Volume 35 (1971), p. 25-27; II. Phys. Lett. D, 5 (1972), 1992-2002 | DOI

[4] Atiyah, M. F.; Patodi, V. K.; Singer, I. M. Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Phil. Soc., Volume 77 (1975), pp. 43-69 | DOI | MR | Zbl

[5] Bollé, D.; Gesztesy, F.; Danneels, C. Threshold scattering in two dimensions, Ann. Inst. H. Poincaré, Sect. A, Volume 48 (1988), pp. 175-204 | EuDML | Numdam | MR | Zbl

[6] Bolley, P.; Camus, J. Opérateurs à indices, II, Séminaire d’analyse fonctionnelle, Publication des séminaires mathématiques, Univ. Rennes 1, 1973 | EuDML | MR

[7] Brüning, J.; Seeley, R. T. An index theorem for first order regular singular operators, Amer. J. Math., Volume 110 (1988), pp. 659-714 | DOI | MR | Zbl

[8] Burq, N.; Planchon, F.; Stalke, J. G.; Shadi Tahvildar-Zadeh, A. Strichartz estimates for the wave and Schrödinger equations with the inverse-square potentials, J. Funct. Analysis, Volume 203 (2003), pp. 519-549 | DOI | MR | Zbl

[9] Cardoso, F.; Vodev, G. Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds, II, Ann. Inst. H. Poincaré, Volume 3 (2002), pp. 673-691 | MR | Zbl

[10] Carron, G. Théorème de l’indice sur les variétés non-compactes, J. reine angew. Math., Volume 541 (2001), pp. 81-115 | DOI | MR | Zbl

[11] Carron, G. A topological criterion for the existence of half-bound states, J. London Math. Soc., Volume 65 (2002), pp. 757-768 | DOI | MR | Zbl

[12] Carron, G. Le saut en zéro de la fonction de décalage spectral, J. Funct. Anal., Volume 212 (2004), pp. 222-260 | DOI | MR | Zbl

[13] Fournais, S.; Skibsted, E. Zero energy asymptotics of the resolvent for a class of slowly decaying potential, Preprint, 2003

[14] Froese, R.; Herbst, I. Exponential bounds and absence of positive eigenvalues for $N$-body Schrödinger operators, Comm. Math. Phys., Volume 87 (1982/83), pp. 429-447 | DOI | MR | Zbl

[15] Hassell, A.; Marcshall, S. Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree $-2$, Preprint, October 2005

[16] Hassell, A.; Wunsch, J. The Schrödinger propagators for scattering metrics, Preprint, 2003

[17] Jensen, A.; Kato, T. Spectral properties of Schrödinger operators and time decay of wave functions, Duke Math. J., Volume 46 (1979), pp. 583-611 | DOI | MR | Zbl

[18] Jensen, A.; Mourre, E.; Perry, P. Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. H. Poincaré, Sect A, Volume 41 (1984), pp. 207-225 | Numdam | MR | Zbl

[19] Jensen, A.; Nenciu, G. A unified approach to resolvent expansions at thresholds, Reviews Math. Phys., Volume 13 (2001), pp. 717-754 | DOI | MR | Zbl

[20] Maz’ya, V.; Nazarov, S.; Plamenevskij, B. Asymptotic Theory of Elliptic Boundary Value Problemes in Singularly Perturbed Domains, 1, Birkhäuser, Boston-Berlin, 2000 | Zbl

[21] Melrose, R. B. The Atiyah-Patodi-Singer index theorem, A. K. Peters Classics, Massachusetts, 1993 | MR | Zbl

[22] Mourre, E. Absence of singular continuous spectrum for certain self-adjoint operators, Comm. Math. Phys., Volume 78 (1981), pp. 391-408 | DOI | MR | Zbl

[23] Müller, W. Relative zeta functions, relative determinants, and scattering theory, Comm. Math. Physics, Volume 192 (1998), pp. 309-347 | DOI | MR | Zbl

[24] Murata, M. Asymptotic expansions in time for solutions of Schrödinger type equations, J. Funct. Analysis, Volume 49 (1982), pp. 10-53 | DOI | MR | Zbl

[25] Nakamura, S. Low energy asymptotics for Schrödinger operators with slowly decreasing potentials, Comm. Math. Phys., Volume 161 (1994), pp. 63-76 | DOI | MR | Zbl

[26] Newton, R. G. Noncentral potentials: the generalized Levinson theorem and the structure of the spectrum, J. Math. Phys., Volume 18 (1977), pp. 1582-1588 | DOI | MR

[27] Newton, R. G. Scattering Theory of Waves and Particles, Springer-Verlag, Berlin, 1982 | MR | Zbl

[28] Olver, F. W. J. Asymptotics and Special Functions, A. K. Peters Classics, Massachusetts, 1997 | MR | Zbl

[29] Rauch, J. Local decay of scattering solutions to Schrödinger’s equation, Comm. Math. Phys., Volume 61 (1978), pp. 149-168 | DOI | MR | Zbl

[30] Robert, D. Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics, J. Funct. Analysis, Volume 126 (1994), pp. 36-82 | DOI | MR | Zbl

[31] Vainberg, B. R. On the short wave asymptotic behaviour of solutions of stationary problems and the asymptoic behaviou as $t\to \infty$ of solutions of non-stationary problems, Russ. Math. Survey, Volume 30 (1975), pp. 1-58 | DOI | MR | Zbl

[32] Vasy, A. Propagation of singularities in three-body scattering, Astérisque, Volume 262 (2000), pp. 6-151 | Numdam | MR | Zbl

[33] Vasy, A.; Zworski, M. Semiclassical estimates in asymptotically Euclidean scattering, Comm. in Math. Phys., Volume 212 (2000), pp. 205-217 | DOI | MR | Zbl

[34] Wang, X. P. Time-decay of scattering solutions and classical trajectories, Ann. Inst. H. Poincaré, Sect A, Volume 47 (1987), pp. 25-37 | Numdam | MR | Zbl

[35] Wang, X. P. Time-decay of scattering solutions and resolvent estimates for semi-classical Schrödinger operators, J. Diff. Equations, Volume 71 (1988), pp. 348-395 | DOI | MR | Zbl

[36] Wang, X. P. Asymptotic behavior of the resolvent of $N$-body Schrödinger operators near a threshold, Ann. Inst. H. Poincaré, Volume 4 (2003), pp. 553-600 | MR | Zbl

[37] Wang, X. P. On the existence of the $N$-body Efimov effect, J. Funct. Analysis, Volume 209 (2004), pp. 137-161 | DOI | MR | Zbl

[38] Wang, X. P. Threshold energy resonance in geometric scattering, Proceedings of Symposium “Scattering and Spectral Theory”, August 2003, Recife, Brazil, Matemática Contemporânea, Volume 26 (2004), pp. 135-164 | MR | Zbl

[39] Watson, G. N. A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[40] Yafaev, D. R. The low-energy scattering for slowly decreasing potentials, Comm. Math. Phys., Volume 85 (1982), pp. 177-198 | DOI | MR | Zbl