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Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians

Patrik Wahlberg


We study propagation of phase space singularities for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of polynomial decay of given order in open cones in the phase space, which gives a parametrized refinement of the Gabor wave front set. The main result confirms the fundamental role of the singular space associated to the quadratic form for the propagation of phase space singularities. The singularities are contained in the singular space, and propagate in the intersection of the singular space and the initial datum singularities along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form.

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Carypis, E. and Wahlberg, P., Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians, J. Fourier Anal. Appl. 23 (2017), no. 3, 530–571.

Cordero, E. and Nicola, F., On the Schrödinger equation with potential in modulation spaces, J. Pseudo-Differ. Oper. Appl. 5 (2014), no. 3, 319–341.

Cordero, E., Nicola, F., and Rodino, L., Propagation of the Gabor wave front set for Schrödinger equations with non-smooth potentials, Rev. Math. Phys. 27 (2015), no. 1, 1550001, 33 pp.

Feichtinger, H. G., Modulation spaces on locally compact Abelian groups, Tech. report, University of Vienna, January 1983; also in: M. Krishna, R. Radha, S. Thangavelu (eds.), Wavelets and their applications, Allied Publishers, New Dehli, 2003, pp. 1-56.

Folland, G. B., Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989.

Gröchenig, K., Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.

Gröchenig, K., Time-frequency analysis of Sjöstrand's class, Rev. Mat. Iberoam. 22 (2006), no. 2, 703–724.

Hitrik, M. and Pravda-Starov, K., Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 344 (2009), no. 4, 801–846.

Hitrik, M. and Pravda-Starov, K., Semiclassical hypoelliptic estimates for non-selfadjoint operators with double characteristics, Comm. Partial Differential Equations 35 (2010), no. 6, 988–1028.

Hitrik, M. and Pravda-Starov, K., Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 3, 985–1032.

Holst, A., Toft, J., and Wahlberg, P., Weyl product algebras and modulation spaces, J. Funct. Anal. 251 (2007), no. 2, 463–491.

Hörmander, L., The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften, vol. 256, Springer-Verlag, Berlin, 1983.

Hörmander, L., Quadratic hyperbolic operators, in “Microlocal analysis and applications (Montecatini Terme, 1989)'', Lecture Notes in Math., vol. 1495, Springer, Berlin, 1991, pp. 118--160.

Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. 219 (1995), no. 3, 413–449.

Nicola, F., Phase space analysis of semilinear parabolic equations, J. Funct. Anal. 267 (2014), no. 3, 727–743.

Nicola, F. and Rodino, L., Propagation of Gabor singularities for semilinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 6, 1715–1732.

Pravda-Starov, K., Contraction semigroups of elliptic quadratic differential operators, Math. Z. 259 (2008), no. 2, 363–391.

Pravda-Starov, K., Subelliptic estimates for quadratic differential operators, Amer. J. Math. 133 (2011), no. 1, 39–89.

Pravda-Starov, K., Rodino, L., and Wahlberg, P., Propagation of gabor singularities for Schrödinger equations with quadratic Hamiltonians, Math. Nachr. 291 (2018), no. 1, 128-159.

Rodino, L. and Wahlberg, P., The Gabor wave front set, Monatsh. Math. 173 (2014), no. 4, 625–655.

Schulz, R. and Wahlberg, P., Microlocal properties of Shubin pseudodifferential and localization operators, J. Pseudo-Differ. Oper. Appl. 7 (2016), no. 1, 91–111.

Shubin, M. A., Pseudodifferential operators and spectral theory, second ed., Springer-Verlag, Berlin, 2001.

Sjöstrand, J., An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994), no. 2, 185–192.

Sjöstrand, J., Wiener type algebras of pseudodifferential operators, in “Séminaire sur les Équations aux Dérivées Partielles, 1994–1995”, Exp. No. IV, École Polytech., Palaiseau, 1995.

Taylor, M. E., Noncommutative harmonic analysis, Mathematical Surveys and Monographs, vol. 22, American Mathematical Society, Providence, RI, 1986.

Viola, J., Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators, Int. Math. Res. Not. IMRN (2013), no. 20, 4615–4671.

Viola, J., Spectral projections and resolvent bounds for partially elliptic quadratic differential operators, J. Pseudo-Differ. Oper. Appl. 4 (2013), no. 2, 145–221.

Yosida, K., Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995.



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