Reconstruction of singularities for solutions of Schrödinger’s equation. (English) Zbl 0554.35031

This paper is concerned with the time-behaviour of singularities of solutions to some Schrödinger equations. There are three types of Hamiltonians considered: \(H=H_ 0+V\), where the perturbation V is a symbol of order 0 (i.e. \(| D_ x^{\alpha}V(x)| \leq c_{\alpha}(1+| x|)^{-| \alpha |})\) and \(H_ 0\) is the Laplacian (the free particle case), the isotropic oscillator or the anisotropic oscillator respectively. In the (perturbed) free particle case it is shown that the kernel \(k_ v(t,x,y)\) of the unitary time evolution operator \(\exp (-itH)\) is smooth in (x,y) for all t. For the isotropic oscillator case, i.e. if \(H_ 0=-1/2\Delta +1/2| x|^ 2\), the singular support of the kernel \(k_ v(t,x,y)\) is determined and it is shown that it decays rapidly in x if \(t=m\pi\), \(m\in {\mathbb{N}}.\)
In the general case of the anisotropic oscillator i.e. if \(H_ 0=- 1/2\Delta +1/2\sum^{n}_{k=1}\omega^ 2_ kx^ 2_ k,\) it is shown that the wave front sets are stable under perturbations V of order 0 in the sense that \(WF(k_ v(t,\cdot,y))\subseteq WF(k(t,\cdot,y))\) where k denotes the kernel of the unperturbed propagator \(\exp (-itH_ 0).\) This is the main result of the paper.
Reviewer: H.Cycon


35J10 Schrödinger operator, Schrödinger equation
47Gxx Integral, integro-differential, and pseudodifferential operators
35A20 Analyticity in context of PDEs
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[1] Asada, K., Fujiwara, D.: On some oscillatory integral transformations inL 2(? n ). Jpn. J. Math.4, 299-361 (1978) · Zbl 0402.44008
[2] Chazarain, J.: Spectre d’un Hamiltonien quantique et m?chanique classique. Commun. PDE Vol. 5, No. 6, 599-644 (1980) · Zbl 0437.70014
[3] Duistemaat, J.: Fourier integral operators New York: Courant Institute of Mathematical Sciences, NYU, 1973
[4] Duistermaat, J.: Oscillatory integrals Lagrange immersions and the unfolding of singularities. Commun. Pure Appl. Math.27, 207-281 (1974) · Zbl 0285.35010
[5] Fujiwara, D.: A construction of the fundamental solution for the Schr?dinger equation. J. Anal. Math.35, 41-96 (1979) · Zbl 0418.35032
[6] Fujiwara, D.: Remarks on convergence of Feynman path integrals. Duke Math. J.47, 559-601 (1980) · Zbl 0457.35026
[7] Guillemin, V., Sternberg, S.: Geometric asymptotics. AMS Math. Surveys14, Providence, Rhode Island: American Mathematical Society 1977 · Zbl 0364.53011
[8] H?rmander, L.: Fourier integral operators I. Acta Math.127, 79-183 (1971) · Zbl 0212.46601
[9] H?rmander, L.: Linear partial differential operators. Berlin, Heidelberg, New York: Springer 1976
[10] Kumanogo, H.: Pseudo-differential operators of multiple symbol and the Calderon-Vaillancourt theorem. J. Math. Soc. Jpn.27, 113-120 (1975) · Zbl 0294.35068
[11] Kumanogo, H.: Hyperbolic systems with diagonal principal part. Commun. P.D.E. Vol.4, 959-1015 (1979) · Zbl 0431.35062
[12] Marsden, J., Weinstein, A.: Book Review of (GS), BAMS Vol.1, 545-533 (1979)
[13] Maslov, V. P.: Theories des perturbations et methodes asymptotiques. Paris: Dunod 1970
[14] Treves, F.: Pseudo differential and Fourier integral operators. New York: Plenum 1981
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