Global parametrices for the Schrödinger propagator and geometric approach to the Hamilton-Jacobi equation | Zendy

Sandro Graffi | Zendy; Lorenzo Zanelli | Zendy

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Global parametrices for the Schrödinger propagator and geometric approach to the Hamilton-Jacobi equation

Author(s) -

Sandro Graffi,

Lorenzo Zanelli

Publication year - 2011

Publication title -

rendiconti lincei matematica e applicazioni

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.824

H-Index - 28

eISSN - 1720-0768

pISSN - 1120-6330

DOI - 10.4171/rlm/585

Subject(s) - semiclassical physics , mathematics , propagator , hamilton–jacobi equation , wkb approximation , hamiltonian (control theory) , mathematical analysis , parametrix , mathematical physics , schrödinger equation , quadratic equation , fourier integral operator , integral equation , quantum mechanics , geometry , physics , quantum , mathematical optimization

— A result is announced concerning a family of semiclassical Fourier Integral Operators representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. The construction is based on the geometrical approach of the corresponding HamiltonJacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.

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