On the spectrum of an anharmonic oscillator

Mughetti, Marco

doi:10.1090/S0002-9947-2014-05896-0

On the spectrum of an anharmonic oscillator
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Trans. Amer. Math. Soc. 367 (2015), 835-865 Request permission

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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