Propagation estimates for \(N\)-body Schrödinger operators. (English) Zbl 0760.35035
This paper proves time decay estimates of the form \([B(t) \exp(-itH) f(H) <x>^{-s'} =\text{O}(t^{-s})]\), where \(H\) is a suitable long-range \(N\)-body Schrödinger operator, \(B(t)\) is a pseudo-differential operator and \(f\) is smooth with compact support. These results imply versions of minimal and large velocity estimates of Sigal and Soffer [I. M. Sigal and A. Soffer, e.g.: Ann. Math., II. Ser. 126, 35-108 (1987; Zbl 0646.47009)], as well as estimates for the “free channel region” which were known in very special cases. The author notes that the results can be extended to time-dependent Hamiltonians. The verification of the author’s hypothesis in concrete examples relies on a partition of unity adapted to the cluster decomposition due to G. M. Graf [Commun. Math. Phys. 132, No. 1, 73-101 (1990; Zbl 0726.35096)].
Reviewer: S.Kichenassamy (Minneapolis)
MSC:
| 35P25 | Scattering theory for PDEs |
| 35Q40 | PDEs in connection with quantum mechanics |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81U05 | \(2\)-body potential quantum scattering theory |
References:
| [1] | [F-H1] Froese, R. G., Herbst, I.: A new proof of the Mourre estimate. Duke Math. J.49 (4), 1075–1085 (1982) · Zbl 0514.35025 · doi:10.1215/S0012-7094-82-04947-X |
| [2] | [F-H2] Froese, R. G., Herbst, I.: Exponential bounds and absence of positive eigenvalues forN-body Schroedinger operators. Commun. Math. Phys.87, 429–447 (1982) · Zbl 0509.35061 · doi:10.1007/BF01206033 |
| [3] | [G] Graf, G. M.: Asymptotic completeness forN-body short-range quantum systems: A new proof ETH-preprint 89-50. Commun. Math. Phys.132, 73–101 (1990) · Zbl 0726.35096 · doi:10.1007/BF02278000 |
| [4] | [H] Hill, R. N.: Proof that theH ion has only one bound state. Details and extension to finite nuclear mass. J. Math. Phys.18, (12), 2316–2330 (1977) · doi:10.1063/1.523241 |
| [5] | [H-S1] Herbst, I., Skibsted, E.: Time-dependent approach to radiation conditions. Duke Math. J.64 (1), (1991) |
| [6] | [H-S2] Herbst, I., Skibsted, E.: in preparation |
| [7] | [I] Isozaki, H.: Differentiability of generalized Fourier transforms associated with Schroedinger operators. J. Math. Kyoto Univ.25 (4), 789–806 (1985) · Zbl 0612.35004 |
| [8] | [J] Jensen, A.: Propagation estimates for Schroedinger-type operators. Trans. Am. Math. Soc.291, (1), (1985) |
| [9] | [J-M-P] Jensen, A., Mourre, E., Perry, P.: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. Henri Poincaré41 (2), 207–225 (1984) · Zbl 0561.47007 |
| [10] | [K] Kitada, H.: Fourier integral operators with weighted symbols and micro-local resolvent estimates. J. Math. Soc. Jpn39 (3), 455–476 (1987) · Zbl 0638.35088 · doi:10.2969/jmsj/03930455 |
| [11] | [M1] Mourre, E.: Operateurs conjugues et proprietes de propagation. Commun. Math. Phys.91, 279–300 (1983) · Zbl 0543.47041 · doi:10.1007/BF01211163 |
| [12] | [M2] Mourre, E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys.78, 391–408 (1981) · Zbl 0489.47010 · doi:10.1007/BF01942331 |
| [13] | [R-S] Reed, M., Simon, B.: Methods of modern mathematical physics II. New York: Academic Press 1975 · Zbl 0308.47002 |
| [14] | [S-S] Sigal, I. M., Soffer, A.: Local decay and propagation estimates for time-dependent and time-independent Hamiltonians. University of Princeton, preprint 88 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
