Applications of a metaplectic calculus to Schrödinger evolutions with non-self-adjoint generators
Joe Viola1
1 Université de Nantes Nantes France
Journées équations aux dérivées partielles (2018), Exposé no. 11, 11 p.

We review the calculus of metaplectic operators and shifts in phase space applied to Gaussian wave packets. Using holomorphic extensions of this calculus, one can reduce the L 2 theory of evolution equations with non-selfadjoint quadratic generators to symplectic linear algebra. We illustrate these methods through an application to the quantum harmonic oscillator with complex perturbation ix.

Publié le :
DOI : 10.5802/jedp.671
Joe Viola 1

1 Université de Nantes Nantes France 
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Joe Viola. Applications of a metaplectic calculus to Schrödinger evolutions with non-self-adjoint generators. Journées équations aux dérivées partielles (2018), Exposé no. 11, 11 p. doi : 10.5802/jedp.671. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.671/

[1] Alexandru Aleman; Joe Viola Singular-value decomposition of solution operators to model evolution equations, Int. Math. Res. Not., Volume 2015 (2015) no. 17, pp. 8275-8288

[2] Alexandru Aleman; Joe Viola On weak and strong solution operators for evolution equations coming from quadratic operators, J. Spectr. Theory, Volume 8 (2018) no. 1, pp. 33-121 | Zbl

[3] Valentine Bargmann On a Hilbert space of analytic functions and an associated integral transform, Commun. Pure Appl. Math., Volume 14 (1961), pp. 187-214

[4] Jean-Michel Bismut A survey of the hypoelliptic Laplacian, Géométrie différentielle, physique mathématique, mathématiques et société. II (Astérisque), Volume 322, Société Mathématique de France, 2008, pp. 39-69 | Zbl

[5] Sébastien Gadat; Laurent Miclo Spectral decompositions and 𝕃 2 -operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, Volume 6 (2013) no. 2, pp. 317-372

[6] Lars Hörmander L 2 estimates for Fourier integral operators with complex phase, Ark. Mat., Volume 21 (1983) no. 2, pp. 283-307

[7] Lars Hörmander The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, 1994 | Zbl

[8] Lars Hörmander Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., Volume 219 (1995) no. 3, pp. 413-449

[9] David Krejčiřík; Petr Siegl; Miloš Tater; Joe Viola Pseudospectra in non-Hermitian quantum mechanics, J. Math. Phys., Volume 53 (2015) no. 10, 103513, 32 pages | Zbl

[10] Jean Leray Lagrangian analysis and quantum mechanics. A mathematical structure related to asymptotic expansions and the Maslov index, MIT Press, 1981 (translated from the French by Carolyn Schroeder) | Zbl

[11] Boris Mityagin; Petr Siegl; Joe Viola Differential operators admitting various rates of spectral projection growth, J. Funct. Anal., Volume 272 (2017) no. 8, pp. 3129-3175

[12] Mona Ben Said; Francis Nier; Joe Viola Quaternionic structure and analysis of some Kramers-Fokker-Planck operators (2018) (https://arxiv.org/abs/1807.01881)

[13] Johannes Sjöstrand Parametrices for pseudodifferential operators with multiple characteristics, Ark. Mat., Volume 12 (1974), pp. 85-130

[14] Johannes Sjöstrand Lectures on resonances (2002) (http://www.math.polytechnique.fr/~sjoestrand/CoursgbgWeb.pdf)

[15] Lloyd N. Trefethen; Mark Embree Spectra and pseudospectra. The behavior of nonnormal matrices and operators, Princeton University Press, 2005 | Zbl

[16] Joe Viola The norm of the non-self-adjoint harmonic oscillator semigroup, Integral Equations Oper. Theory, Volume 85 (2016) no. 4, pp. 513-538

[17] Joe Viola The elliptic evolution of non-self-adjoint degree-2 Hamiltonians (2017) (https://arxiv.org/abs/1701.00801)

[18] Maciej Zworski Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012

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