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Dispersion of Singularities of Solutions for Schrödinger Equations

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Abstract

We consider the singularities of solutions for the Schrödinger evolution equation associated with where Q is a d×d real symmetric matrix with the eigenvalues λ1,⋯,λ d , and W ∈ C∞(Rd,R) satisfies W(x)=o(|x|2) as |x|→∞. Under additional conditions, we show the dispersion of microlocal singularities of solutions due to the principal symbol in all directions at time and in the nondegenerate directions at t ∈ Σ. We also show the weaker dispersion of microlocal singularities of solutions due to the subprincipal symbol W in the degenerate directions at t ∈ Σ if W satisfies W(x)=O(|x|1+δδ) as |x|→∞ for some 0<δ<1 and additional conditions. In particular, we prove the dispersion of microlocal singularities of solutions at resonant times when H is a perturbed harmonic oscillator.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka, 560-0043, Japan

    Shin-ichi Doi

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  1. Shin-ichi Doi
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Correspondence to Shin-ichi Doi.

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Communicated by B. Simon

Partly supported by Grand-in-Aid for Young Scientists (B) 14740110, Japan Society of the Promotion of Science; and Mathematical Sciences Research Institute in Berkeley

Dedicated to Professor Mitsuru Ikawa on his sixtieth birthday

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Doi, Si. Dispersion of Singularities of Solutions for Schrödinger Equations. Commun. Math. Phys. 250, 473–505 (2004). https://doi.org/10.1007/s00220-004-1086-7

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  • DOI: https://doi.org/10.1007/s00220-004-1086-7

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