NDLI: Semiclassical Low Energy Scattering for One-Dimensional Schrödinger Operators with Exponentially Decaying Potentials

Content Provider	Springer Nature Link
Author	Costin, Ovidiu Donninger, Roland Schlag, Wilhelm Tanveer, Saleh
Copyright Year	2011
Abstract	We consider semiclassical Schrödinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{{\rm d}^2}{{\rm d}x^2}+V(\cdot;\hbar)$$ with $${\hbar >0 }$$ small. The potential V is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions $${f_\pm(\cdot,E;\hbar)}$$ with error terms that are uniformly controlled for small E and $${\hbar}$$ , and construct the scattering matrix as well as the semiclassical spectral measure associated with $${H(\hbar)}$$ . This is crucial in order to obtain decay bounds for the corresponding wave and Schrödinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta ℓ where the role of the small parameter $${\hbar}$$ is played by ℓ −1. It follows from the results in this paper and Donninger et al. (Commun Math Phys 2009, arXiv:0911.3179), that the decay bounds obtained in Donninger et al. (Adv Math 226(1):484–540, 2011) and Donninger and Wilhelm (Int Math Res Not IMRN 22:4276–4300, 2010) for individual angular momenta ℓ can be summed to yield the sharp t −3 decay for data without symmetry assumptions.
Starting Page	1371
Ending Page	1426
Page Count	56
File Format	PDF
ISSN	14240637
Journal	Annales Henri Poincaré
Volume Number	13
Issue Number	6
e-ISSN	14240661
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2011-12-29
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical Methods in Physics Dynamical Systems and Ergodic Theory Elementary Particles, Quantum Field Theory Classical and Quantum Gravitation, Relativity Theory Theoretical, Mathematical and Computational Physics Quantum Physics
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Nuclear and High Energy Physics Mathematical Physics

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
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3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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