Well-posedness for degenerate Schrödinger equations

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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

Abstract
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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 and 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.
[2]	A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation, Rend. Semin. Mat. Torino, 67 (2009), 151-164.
[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.
[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.
[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.
[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.
[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.
[8]	W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math., 21 (1984), 565-581.
[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.
[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.
[11]	W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620.
[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.
[13]	K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations, Bull. Sci. Math., 119 (1995), 459-473.
[14]	K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.
[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.

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.
[2]	A. Ascanelli and M. Cicognani, Gevrey solutions for a vibrating beam equation, Rend. Semin. Mat. Torino, 67 (2009), 151-164.
[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.
[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.
[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.
[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.
[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.
[8]	W. Ichinose, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math., 21 (1984), 565-581.
[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.
[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.
[11]	W. Ichinose, On the Cauchy problem for Schrödinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620.
[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.
[13]	K. Kajitani and A. Baba, The Cauchy problem for Schrödinger type equations, Bull. Sci. Math., 119 (1995), 459-473.
[14]	K. Kajitani and T. Nishitani, The Hyperbolic Cauchy Problem, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991.
[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.

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