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- W4298053119 abstract "We discuss spectral properties of the family of quartic oscillators $mathfrak h_{mathcal M}(alpha) =-frac{d^2}{dt^2} +Big(frac{1}{2} t^{2} -alphaBig)^2$ on the real line, where $alphain mathbb{R}$ is a parameter. This operator appears in a variety of applications coming from quantum mechanics to harmonic analysis on Lie groups, Riemannian geometry and superconductivity. We study the variations of the eigenvalues $lambda_j(alpha)$ of $mathfrak h_{mathcal M}(alpha)$ as functions of the parameter $alpha$.We prove that for $j$ sufficiently large, $alpha mapsto lambda_j(alpha)$ has a unique critical point, which is a nondegenerate minimum.We also prove that the first eigenvalue $lambda_1(alpha)$ enjoys the same property and give a numerically assisted proof that the same holds for the second eigenvalue $lambda_2(alpha)$. The proof for excited states relies on a semiclassical reformulation of the problem. In particular, we develop a method permitting to differentiate with respect to the semiclassical parameter, which may be of independent interest." @default.
- W4298053119 created "2022-10-01" @default.
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- W4298053119 date "2022-09-27" @default.
- W4298053119 modified "2023-09-26" @default.
- W4298053119 title "On critical points of eigenvalues of the Montgomery family of quartic oscillators" @default.
- W4298053119 hasPublicationYear "2022" @default.
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