Abstract
We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin’s class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.
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Asada, K., Fujiwara, D.: On some oscillatory integral transformations in \(L^2(\mathbb{R}^n)\). Jpn J. Math. 4(2), 299–361 (1978)
Boggiatto, P., Buzano, E., Rodino, L.: Global Hypoellipticity and Spectral Theory, Mathematical Research, vol. 92. Akademie Verlag, Berlin (1996)
Cappiello, M., Gramchev, T., Rodino, L.: Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients. J. Funct. Anal. 237, 634–654 (2006)
Cappiello, M., Nicola, F.: Regularity and decay of solutions of nonlinear harmonic oscillators. Adv. Math. 229, 1266–1299 (2012)
Cappiello, M., Rodino, L., Toft, J.: On the inverse to the harmonic oscillator. Commun. Partial Differ. Equ. 40(6), 1096–1118 (2015)
Cappiello, M., Schulz, R., Wahlberg, P.: Conormal distributions in the Shubin calculus of pseudodifferential operators. J. Math. Phys. 59(2), 021502 (2018)
Cappiello, M., Schulz, R., Wahlberg, P.: Lagrangian distributions and Fourier integral operators with quadratic phase functions and Shubin amplitudes (2018), arXiv:1802.04729
Cordero, E., Gröchenig, K., Nicola, F., Rodino, L.: Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class. J. Math. Phys. 55(8), 081506 (2014)
Cordero, E., Nicola, F., Rodino, L.: Integral representation for the class of generalized metaplectic operators. J. Fourier Anal. Appl. 21, 694–714 (2015)
Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, vol. 194. Springer, Berlin (2000)
de Gosson, M.A.: Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Pseudo-Differential Operators, vol. 7. Theory and Applications, Birkhäuser, Basel (2011)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)
Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques. Astérisque 112, (1984)
Helffer, B., Robert, D.: Comportement asymptotique précisé du spectre d’opérateurs globalement elliptiques dans \(\mathbb{R}^n\). Séminaire équations aux dérivées partielles (Polytechnique), exp. no 2, pp. 1–22 (1980–1981)
Holst, A., Toft, J., Wahlberg, P.: Weyl product algebras and modulation spaces. J. Funct. Anal. 251, 463–491 (2007)
Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I.III,IV. Springer, Berlin (1990)
Hörmander, L.: Quadratic hyperbolic operators. In: Cattabriga, L., Rodino, L. (eds.) Microlocal Analysis and Applications (Montecatini Terme, 1989). Lecture Notes in Mathematics, vol. 1495, pp. 118–160. Springer, Berlin (1991)
Hörmander, L.: Symplectic classification of quadratic forms, and general Mehler formulas. Math. Z. 219(3), 413–449 (1995)
Leray, J.: Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index. The MIT Press, Cambridge (1981)
Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces. Birkhäuser, Basel (2010)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44. Springer, New York (1983)
Pravda-Starov, K., Rodino, L., Wahlberg, P.: Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians. Math. Nachr. 291(1), 128–159 (2018)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1. Academic Press, New York (1980)
Rodino, L., Wahlberg, P.: The Gabor wave front set. Monaths. Math. 173(4), 625–655 (2014)
Schulz, R., Wahlberg, P.: Microlocal properties of Shubin pseudodifferential and localization operators. J. Pseudo-Differ. Oper. Appl. 7(1), 91–111 (2015)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (2001)
Tataru, D.: Phase space transforms and microlocal analysis. Phase space analysis of partial differential equations, Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, pp. 505–524 (1991)
Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)
Weinstein, A.: A symbol class for some Schrödinger equations on \(\mathbb{R}^n\). Am. J. Math. 107(1), 1–21 (1985)
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R. Schulz gratefully acknowledges support of the project “Fourier Integral Operators, symplectic geometry and analysis on noncompact manifolds” received by the University of Turin in form of an “I@Unito” fellowship as well as institutional support by the University of Hannover.
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Cappiello, M., Schulz, R. & Wahlberg, P. Shubin type Fourier integral operators and evolution equations. J. Pseudo-Differ. Oper. Appl. 11, 119–139 (2020). https://doi.org/10.1007/s11868-019-00288-0
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DOI: https://doi.org/10.1007/s11868-019-00288-0