American Institute of Mathematical Sciences

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March  2014, 3(1): 15-33. doi: 10.3934/eect.2014.3.15

Well-posedness for degenerate Schrödinger equations

Massimo Cicognani 1, and  Michael Reissig 2,

1. 

Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40126 Bologna, Italy
2. 

TU Bergakademie Freiberg, Fakultät für Mathematik und Informatik, Institut für Angewandte Analysis, 09596 Freiberg, Germany

Received  March 2013 Revised  August 2013 Published  February 2014

We consider the initial value problem for Schrödinger type equations $$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.
Keywords: Gevrey well-posedness., Schrödinger equations, Cauchy problem, time-depending Hamiltonian, Levi conditions.
Mathematics Subject Classification: Primary: 35J10; Secondary: 35Q4.
Citation: Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
References:
[1]

A. Ascanelli, M. Cicognani and F. Colombini, The global Cauchy problem for a vibrating beam equation, Journal of Differential Equations, 247 (2009), 1440-1451. doi: 10.1016/j.jde.2009.06.012.  Google Scholar

[2]

A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation, Rend. Semin. Mat. Torino, 67 (2009), 151-164.  Google Scholar

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M. Cicognani and F. Colombini, Optimal well-posedness of the cauchy problem for evolution equations with $C^N$ coefficients, Differential and Integral Equations, 17 (2004), 1079-1092.  Google Scholar

[4]

M. Cicognani and T. Herrmanni, $H^\infty$ well-posedness for a $2$-evolution Cauchy problem with complex coefficients, Journal of Pseudo-Differential Operators and Applications, 4 (2013), 63-90. doi: 10.1007/s11868-013-0062-4.  Google Scholar

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S. I. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328.  Google Scholar

[6]

S. I. Doi, Remarks on the Cauchy problem for Schrödingerr-type equations, Comm. Partial Differential Equations, 21 (1996), 163-178. doi: 10.1080/03605309608821178.  Google Scholar

[7]

M. Dreher, Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces, Bull. Sci. Math., 127 (2003), 485-503. doi: 10.1016/S0007-4497(03)00026-5.  Google Scholar

[8]

W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math., 21 (1984), 565-581.  Google Scholar

[9]

W. Ichinose, Sufficient condition on $H^\infty$ well-posedness for Schrödinger type equations, Comm. Partial Differential Equations, 9 (1984), 33-48. doi: 10.1080/03605308408820324.  Google Scholar

[10]

W. Ichinose, On a necessary condition for $L^2$ well-posedness of the Cauchy problem for some Schrödinger type equations with a potential term, J. Math. Kyoto Univ., 33 (1993), 647-663.  Google Scholar

[11]

W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620.  Google Scholar

[12]

K. Kajitani, The Cauchy problem for Schrödinger type equations with variable coefficients, J. Math. Soc. Japan, 50 (1998), 179-202. doi: 10.2969/jmsj/05010179.  Google Scholar

[13]

K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations, Bull. Sci. Math., 119 (1995), 459-473.  Google Scholar

[14]

K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.  Google Scholar

[15]

S. Mizohata, On some Schrödinger type equations, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 81-84. doi: 10.3792/pjaa.57.81.  Google Scholar

show all references

References:
[1]

A. Ascanelli, M. Cicognani and F. Colombini, The global Cauchy problem for a vibrating beam equation, Journal of Differential Equations, 247 (2009), 1440-1451. doi: 10.1016/j.jde.2009.06.012.  Google Scholar

[2]

A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation, Rend. Semin. Mat. Torino, 67 (2009), 151-164.  Google Scholar

[3]

M. Cicognani and F. Colombini, Optimal well-posedness of the cauchy problem for evolution equations with $C^N$ coefficients, Differential and Integral Equations, 17 (2004), 1079-1092.  Google Scholar

[4]

M. Cicognani and T. Herrmanni, $H^\infty$ well-posedness for a $2$-evolution Cauchy problem with complex coefficients, Journal of Pseudo-Differential Operators and Applications, 4 (2013), 63-90. doi: 10.1007/s11868-013-0062-4.  Google Scholar

[5]

S. I. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328.  Google Scholar

[6]

S. I. Doi, Remarks on the Cauchy problem for Schrödingerr-type equations, Comm. Partial Differential Equations, 21 (1996), 163-178. doi: 10.1080/03605309608821178.  Google Scholar

[7]

M. Dreher, Necessary conditions for the well-posedness of Schrödinger type equations in Gevrey spaces, Bull. Sci. Math., 127 (2003), 485-503. doi: 10.1016/S0007-4497(03)00026-5.  Google Scholar

[8]

W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math., 21 (1984), 565-581.  Google Scholar

[9]

W. Ichinose, Sufficient condition on $H^\infty$ well-posedness for Schrödinger type equations, Comm. Partial Differential Equations, 9 (1984), 33-48. doi: 10.1080/03605308408820324.  Google Scholar

[10]

W. Ichinose, On a necessary condition for $L^2$ well-posedness of the Cauchy problem for some Schrödinger type equations with a potential term, J. Math. Kyoto Univ., 33 (1993), 647-663.  Google Scholar

[11]

W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620.  Google Scholar

[12]

K. Kajitani, The Cauchy problem for Schrödinger type equations with variable coefficients, J. Math. Soc. Japan, 50 (1998), 179-202. doi: 10.2969/jmsj/05010179.  Google Scholar

[13]

K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations, Bull. Sci. Math., 119 (1995), 459-473.  Google Scholar

[14]

K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.  Google Scholar

[15]

S. Mizohata, On some Schrödinger type equations, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 81-84. doi: 10.3792/pjaa.57.81.  Google Scholar

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