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Jensen, Arne; Mourre, Éric; Perry, Peter

Multiple commutator estimates and resolvent smoothness in quantum scattering theory. (English) Zbl 0561.47007

Ann. Inst. Henri Poincaré, Phys. Théor. 41, 207-225 (1984).
Let H be a selfadjoint operator with domain \({\mathcal D}(H)\) in a Hilbert space \({\mathcal H}\). H is said to admit A as a conjugate operator at \(E\in R\), if A is a selfadjoint operator satisfying certain domain conditions, and if, furthermore, the following operator estimate holds: \[ (*)\quad E_ H(J)i[H,A]E_ H(J)\geq \alpha E_ H(J)+K. \] Here \(E_ H\) denotes the spectral measure of H, \(J=(E-\delta\), \(E+\delta)\), \(\delta >0\), \(\alpha >0\), and K is compact. One furthermore assumes that the multiple commutators \(i[i[H,A],A]...\) up to order n define bounded operators from \({\mathcal D}(H)\) (graph norm) to \({\mathcal H}\). A weaker condition is also needed on the \((n+1)\)-fold commutator.
The main results are that the boundary values \(R(\alpha \pm i0)=\lim_{\epsilon \downarrow 0}(H-\lambda \mp i\epsilon)^{-1}\) exist and are \(C^ n\) w.r.t. \(\lambda\) in suitable spaces defined using A. Here \(\lambda \in J\setminus \sigma_ p(H),\) where \(\sigma_ p(H)\) denotes the point spectrum of H. Under condition (*) \(\sigma_ p(H)\) is known to be discrete in J. Detailed results are too complicated to be stated here.
The applications include an abstract scattering theory using a conjugate operator, and a proof of completeness of modified wave operators for certain Schrödinger operators with long range potentials.

Cited in 2 Reviews
Cited in 67 Documents

MSC:

47A40 Scattering theory of linear operators
47A10 Spectrum, resolvent
81U05 \(2\)-body potential quantum scattering theory
47F05 General theory of partial differential operators
47B47 Commutators, derivations, elementary operators, etc.

Keywords:

selfadjoint operator; conjugate operator; spectral measure; multiple commutators; abstract scattering theory; completeness of modified wave operators for certain Schrödinger operators with long range potentials
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References:

[1] S. Agmon , Some new results in spectral and scattering theory of differential operators on L2(Rn) . Séminaire Goulaouic-Schwartz , 1978/1979 . Exp. No. 2, École Polytech ., Palaiseau , 1979 . Numdam | MR 557513 | Zbl 0406.35052 · Zbl 0406.35052
[2] J. Aguilar , J.M. Combes , Comm. Math. Phys. , t. 22 , 1971 , p. 269 - 279 . Article | MR 345551 | Zbl 0219.47011 · Zbl 0219.47011 · doi:10.1007/BF01877510
[3] E. Balslev , J.M. Combes , Comm. Math. Phys. , t. 22 , 1971 , p. 280 - 294 . Article | MR 345552 | Zbl 0219.47005 · Zbl 0219.47005 · doi:10.1007/BF01877511
[4] J.D. Dollard , J. Math. Phys. , t. 5 , 1964 , p. 729 - 738 . MR 163620
[5] V. Enss , Ann. Phys. ( N. Y .), t. 119 , 1979 , p. 117 - 132 . Addendum , Preprint, Bielefeld , 1979 . MR 535624
[6] L. Hormander , Math. Z. , t. 146 , 1976 , p. 69 - 91 . Article | MR 393884 | Zbl 0319.35059 · Zbl 0319.35059 · doi:10.1007/BF01213717
[7] T. Ikebe , J. Funct. Anal. , t. 20 , 1975 , p. 158 - 177 . Zbl 0315.35067 · Zbl 0315.35067 · doi:10.1016/0022-1236(75)90048-8
[8] T. Ikebe , Publ. Res. Inst. Math. Sci. , t. 11 , 1976 , p. 551 - 558 . MR 440213 | Zbl 0345.35032 · Zbl 0345.35032 · doi:10.2977/prims/1195191477
[9] T. Ikebe , H. Isozaki , Publ. Res. Inst. Math. Sci. , t. 15 , 1979 , p. 679 - 718 . MR 566076 | Zbl 0432.35061 · Zbl 0432.35061 · doi:10.2977/prims/1195187871
[10] T. Ikebe , H. Isozaki , Integral Equations Operator Theory , t. 5 , 1982 , p. 18 - 49 . MR 646878
[11] H. Isozaki , Publ. Res. Inst. Math. Sci. , t. 13 , 1977 , p. 589 - 626 . MR 479087 | Zbl 0435.47011 · Zbl 0435.47011 · doi:10.2977/prims/1195189601
[12] H. Isozaki , H. Kitada , Private Communication .
[13] A. Jensen , T. Kato , Duke Math. J. , t. 46 , 1979 , p. 583 - 611 . Article | MR 544248 | Zbl 0448.35080 · Zbl 0448.35080 · doi:10.1215/S0012-7094-79-04631-3
[14] A. Jensen , Comm. Math. Phys. , t. 82 , 1981 , p. 435 - 456 . Article | MR 641772 | Zbl 0483.47031 · Zbl 0483.47031 · doi:10.1007/BF01237049
[15] A. Jensen , Lett. Math. Phys. , t. 7 , 1983 , p. 137 - 143 . MR 708436 | Zbl 0517.47010 · Zbl 0517.47010 · doi:10.1007/BF00419932
[16] T. Kato , Math. Ann. , t. 162 , 1966 , p. 258 - 279 . MR 190801 | Zbl 0139.31203 · Zbl 0139.31203 · doi:10.1007/BF01360915
[17] T. Kato , Studia Math. , t. 31 , 1968 , p. 535 - 546 . MR 234314 | Zbl 0215.48802 · Zbl 0215.48802
[18] H. Kitada , Proc. Japan Acad. Ser. A Math. Sci. , t. 52 , 1976 , p. 409 - 412 . Article | MR 422911 | Zbl 0358.35024 · Zbl 0358.35024 · doi:10.3792/pja/1195518239
[19] H. Kitada , J. Math. Soc. Japan , t. 29 , 1977 , p. 665 - 691 . Article | MR 634802 | Zbl 0356.47006 · Zbl 0356.47006 · doi:10.2969/jmsj/02940665
[20] H. Kitada , J. Math. Soc. Japan , t. 30 , 1978 , p. 603 - 632 . Article | MR 634803 | Zbl 0388.35055 · Zbl 0388.35055 · doi:10.2969/jmsj/03040603
[21] R. Lavine , Comm. Math. Phys. , t. 20 , 1971 , p. 301 - 323 . Article | MR 293945 | Zbl 0207.13706 · Zbl 0207.13706 · doi:10.1007/BF01646626
[22] R. Lavine , J. Math. Phys. , t. 14 , 1973 , p. 376 - 379 . Zbl 0269.47005 · Zbl 0269.47005 · doi:10.1063/1.1666324
[23] E. Mourre , Comm. Math. Phys. , t. 68 , 1979 , p. 91 - 94 . Article | MR 539739 | Zbl 0429.47006 · Zbl 0429.47006 · doi:10.1007/BF01562544
[24] E. Mourre , Comm. Math. Phys. , t. 78 , 1981 , p. 391 - 408 . Article | MR 603501 | Zbl 0489.47010 · Zbl 0489.47010 · doi:10.1007/BF01942331
[25] E. Mourre , Algebraic approach to some propagation properties of the Schrödinger equation . In: Mathematical Problems of Theoretical Physics , ed. R. Schrader, R. Seiler, P. A. Uhlenbrock. Lecture Notes in Physics , No. 153 , Springer Verlag , Berlin , etc., 1981 . · Zbl 0489.47010
[26] E. Mourre , Comm. Math. Phys. , t. 91 , 1983 , p. 279 - 300 . Article | MR 723552 | Zbl 0543.47041 · Zbl 0543.47041 · doi:10.1007/BF01211163
[27] E. Mourre , Preprint, CPT-CNRS , Marseille , 1982 .
[28] P.L. Muthuramalingam , K.B. Sinha , Preprint, Indian Statistical Institute , 1983 .
[29] P. Perry , Comm. Math. Phys. , t. 81 , 1981 , p. 243 - 259 . Article | MR 632760 | Zbl 0471.47007 · Zbl 0471.47007 · doi:10.1007/BF01208898
[30] P. Perry , I. Sigal , B. Simon , Ann. Math. , t. 114 , 1981 , p. 519 - 567 . MR 634428 | Zbl 0477.35069 · Zbl 0477.35069 · doi:10.2307/1971301
[31] C.R. Putnam , Commutation Properties of Hilbert Space Operators and Related Topics . Springer Verlag , Berlin , etc., 1967 . MR 217618 | Zbl 0149.35104 · Zbl 0149.35104
[32] M. Reed , B. Simon , Methods of Modern Mathematical Physics. III. Scattering Theory . Academic Press , New York , 1979 . MR 529429 | Zbl 0405.47007 · Zbl 0405.47007
[33] Y. Saito , Osaka J. Math. , t. 14 , 1979 , p. 37 - 53 . MR 442505 | Zbl 0356.35020 · Zbl 0356.35020
[34] Y. Saito , Spectral representations for Schrödinger operators with long-range potentials . Lecture Notes in Mathematics , Vol. 727 , Springer Verlag , Berlin , etc., 1979 . MR 540891 | Zbl 0414.47012 · Zbl 0414.47012
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