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On the spectrum of an anharmonic oscillator


Author: Marco Mughetti
Journal: Trans. Amer. Math. Soc. 367 (2015), 835-865
MSC (2010): Primary 34L15
DOI: https://doi.org/10.1090/S0002-9947-2014-05896-0
Published electronically: October 16, 2014
MathSciNet review: 3280029
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Abstract: We consider the one-dimensional anharmonic oscillator $ -\partial ^2_x+ax^{2h}+\mu x^{h-1}$ and we study the qualitative behaviour of its eigenvalues $ \lambda _j$; in particular, we show how the sign of its eigenvalues depends on the parameters $ h\in \mathbb{N},$ $ a\in \mathbb{R}_+,$ $ \mu \in \mathbb{R}$. We applied our results to the study of $ C^\infty $-hypoellipticity and of a priori estimates for certain non-transversally elliptic operators.


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Additional Information

Marco Mughetti
Affiliation: Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy
Email: mughetti@dm.unibo.it

DOI: https://doi.org/10.1090/S0002-9947-2014-05896-0
Keywords: Anharmonic oscillator, pseudodifferential operators, eigenvalues, a priori estimates, hypoellipticity
Received by editor(s): December 29, 2011
Received by editor(s) in revised form: June 15, 2012
Published electronically: October 16, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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