Conditional stability for a Cauchy problem for the ultrahyperbolic Schrödinger equation

ABSTRACT

In this work, we first establish a local Carleman estimate for the ultrahyperbolic Schrödinger equation which arises in some theories of modern physics such as quantum mechanics and string theory. Next, we prove a Hölder stability estimate for the Cauchy Problem. In the proof, we introduce a suitable cut-off function and extend Cauchy data in a Sobolev space to reduce the problem to another problem for functions with compact supports and then we apply the Carleman estimate.

KEYWORDS:

• Ultrahyperbolic Schrödinger equation
• Cauchy problem
• conditional stability
• Carleman inequality

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35A23
• 76B15
• 35B35

Acknowledgments

The authors would like to thank Professor Masahiro Yamamoto from The University of Tokyo for his very valuable comments on the problem. They would also like to thank the anonymous referees for their constructive feedback.

Disclosure statement

No potential conflict of interest was reported by the author(s).