The trace of the generalized harmonic oscillator
Annales de l'Institut Fourier, Volume 49 (1999) no. 1, p. 351-373

We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator

$\left({D}_{t}+\frac{1}{2}\Delta +V\right)\psi =0\phantom{\rule{2em}{0ex}}\left(0.1\right)$

where $\Delta$ is the Laplace-Beltrami operator with respect to a “scattering metric” on a compact manifold $M$ with boundary (the class of scattering metrics is a generalization of asymptotically Euclidean metrics on ${ℝ}^{n}$, radially compactified to the ball) and $V$ is a perturbation of $\frac{1}{2}{\omega }^{2}{x}^{-2}$, with $x$ a boundary defining function for $M$ (e.g. $x=1/r$ in the compactified Euclidean case). Using the quadratic-scattering wavefront set, a generalization of Hörmander’s wavefront set that measures oscillation at $\partial M$ as well as singularities, we describe a propagation of singularities theorem for solutions of (0.1). This enables us to prove the following trace theorem : let

${S}_{\omega }=\left\{\frac{L}{\omega }:\text{there}\phantom{\rule{4pt}{0ex}}\text{exists}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{closed}\phantom{\rule{4pt}{0ex}}\text{geodesic}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{3.33333pt}{0ex}}\partial M\phantom{\rule{3.33333pt}{0ex}}\text{of}\phantom{\rule{4pt}{0ex}}\text{length}\phantom{\rule{3.33333pt}{0ex}}±L\right\}$$\cup \left\{\frac{n\pi }{\omega }:\text{there}\phantom{\rule{4pt}{0ex}}\text{exists}\phantom{\rule{4pt}{0ex}}\text{a}\phantom{\rule{4pt}{0ex}}\text{geodesic}\phantom{\rule{3.33333pt}{0ex}}n\text{-gon}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{3.33333pt}{0ex}}M\phantom{\rule{3.33333pt}{0ex}}\text{with}\phantom{\rule{4pt}{0ex}}\text{vertices}\phantom{\rule{3.33333pt}{0ex}}\partial M\right\}\cup \left\{0\right\}.$

Let $U\left(t\right)={\mathrm{e}}^{it\left(\frac{1}{2}\Delta +V\right)}$ be the solution operator to the Cauchy problem for (0.1). Then under a non-trapping assumption for the geodesic flow on $\stackrel{\circ }{\phantom{\rule{0.0pt}{0ex}}M}$, we have

$\mathrm{supp}\phantom{\rule{3.33333pt}{0ex}}\mathrm{sing}\phantom{\rule{3.33333pt}{0ex}}\mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}U\left(t\right)\subset {S}_{\omega },$

where $\mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}U\left(t\right)$ is the distribution given by integrating the Schwartz kernel of $U\left(t\right)$ over the diagonal in $M×M$ or, alternatively, by ${\sum }_{j}{\mathrm{e}}^{-it{\lambda }_{j}}$, where ${\lambda }_{j}$ are the eigenvalues of $\frac{1}{2}\Delta +V$.

On étudie une généralisation géométrique de l’équation de Schrödinger dépendant du temps pour l’oscillateur harmonique

$\left({D}_{t}+\frac{1}{2}\Delta +V\right)\psi =0,\phantom{\rule{2em}{0ex}}\left(0.1\right)$

$\Delta$ est l’opérateur de Laplace-Beltrami associé à une “métrique scattering” sur une variété compacte $M$ à bord (la classe des métriques scattering est une généralisation des métriques asymptotiquement euclidiennes sur ${ℝ}^{n}$, compactifié radialement en une boule) et $V$ est une perturbation de $\frac{1}{2}{\omega }^{2}{x}^{-2}$, où $x$ est une fonction qui définit le bord de $M$ ( par exemple, $x=1/r$ dans le cas euclidien compactifié). En employant le front d’onde quadratique-scattering, une généralisation du front d’onde de Hörmander qui mesure les oscillations sur $\partial M$ ainsi que les singularités, on décrit un théorème de propagation des singularités pour les solutions de (0.1). Ceci permet de démontrer le théorème de trace suivant : soit

${S}_{\omega }=\left\{\frac{L}{\omega };\phantom{\rule{3.33333pt}{0ex}}\text{il}\phantom{\rule{4pt}{0ex}}\text{existe}\phantom{\rule{4pt}{0ex}}\text{une}\phantom{\rule{4pt}{0ex}}\text{géodésique}\phantom{\rule{4pt}{0ex}}\text{fermée}\phantom{\rule{4pt}{0ex}}\text{dans}\phantom{\rule{3.33333pt}{0ex}}\partial M\phantom{\rule{3.33333pt}{0ex}}\text{de}\phantom{\rule{4pt}{0ex}}\text{longueur}\phantom{\rule{3.33333pt}{0ex}}±L\right\}\cup$$\left\{\frac{n\pi }{\omega };\phantom{\rule{3.33333pt}{0ex}}\text{il}\phantom{\rule{4pt}{0ex}}\text{existe}\phantom{\rule{4pt}{0ex}}\text{un}\phantom{\rule{4pt}{0ex}}n\text{-polygone}\phantom{\rule{4pt}{0ex}}\text{géodésique}\phantom{\rule{4pt}{0ex}}\text{dans}\phantom{\rule{3.33333pt}{0ex}}M,\phantom{\rule{3.33333pt}{0ex}}\text{à}\phantom{\rule{4pt}{0ex}}\text{sommets}\phantom{\rule{4pt}{0ex}}\text{sur}\phantom{\rule{3.33333pt}{0ex}}\partial M\right\}\cup \left\{0\right\}.$

Soit $U\left(t\right)={\mathrm{e}}^{it\left(\frac{1}{2}\Delta +V\right)}$ l’opérateur solution du problème de Cauchy pour (0.1). Alors sous une hypothèse de non-captivité pour le flot géodésique sur $\stackrel{\circ }{\phantom{\rule{0.0pt}{0ex}}M}$, on a

$\mathrm{supp}\phantom{\rule{3.33333pt}{0ex}}\mathrm{sing}\phantom{\rule{3.33333pt}{0ex}}\mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}U\left(t\right)\subset {S}_{\omega },$

$\mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}U\left(t\right)$ est la distribution qu’on obtient en intégrant le noyau de Schwartz de $U\left(t\right)$ sur la diagonale de $M×M$, ou bien en prenant ${\sum }_{j}{\mathrm{e}}^{-it{\lambda }_{j}}$, où les ${\lambda }_{j}$ sont les valeurs propres de $\frac{1}{2}\Delta +V$.

@article{AIF_1999__49_1_351_0,
author = {Wunsch, Jared},
title = {The trace of the generalized harmonic oscillator},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {49},
number = {1},
year = {1999},
pages = {351-373},
doi = {10.5802/aif.1677},
zbl = {0924.58098},
mrnumber = {2000c:58055},
language = {en},
url = {http://www.numdam.org/item/AIF_1999__49_1_351_0}
}

Wunsch, Jared. The trace of the generalized harmonic oscillator. Annales de l'Institut Fourier, Volume 49 (1999) no. 1, pp. 351-373. doi : 10.5802/aif.1677. http://www.numdam.org/item/AIF_1999__49_1_351_0/

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