## 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

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 47Gxx Integral, integro-differential, and pseudodifferential operators 35A20 Analyticity in context of PDEs
Full Text:

### References:

 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.