The trace of the generalized harmonic oscillator

Jared Wunsch

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 1, page 351-373
  • ISSN: 0373-0956

Abstract

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We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator D t + 1 2 Δ + V ψ = 0 ( 0 . 1 ) where Δ 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 1 2 ω 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 ∂ 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 ω = L ω : there exists a closed geodesic in ∂ M of length ± L ∪ n π ω : there exists a geodesic n -gon in M with vertices ∂ M ∪ { 0 } . Let U ( t ) = e i t ( 1 2 Δ + V ) be the solution operator to the Cauchy problem for (0.1). Then under a non-trapping assumption for the geodesic flow on M ∘ , we have supp sing Tr U ( t ) ⊂ S ω , where Tr U ( t ) is the distribution given by integrating the Schwartz kernel of U ( t ) over the diagonal in M × M or, alternatively, by ∑ j e - i t λ j , where λ j are the eigenvalues of 1 2 Δ + V .

How to cite

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Wunsch, Jared. "The trace of the generalized harmonic oscillator." Annales de l'institut Fourier 49.1 (1999): 351-373. <http://eudml.org/doc/75340>.

@article{Wunsch1999,
abstract = {We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator\begin\{\}\Big (D\_t+\{1\over 2\}\Delta +V\Big )\psi =0\qquad (0.1)\end\{\}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 $\{\Bbb R\}^n$, radially compactified to the ball) and $V$ is a perturbation of $\{1\over 2\}\omega ^2x^\{-2\}$, 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\begin\{\}S\_\omega =\Big \lbrace \{L\over \omega \}:\text\{there\} \text\{exists\} \text\{a\} \text\{closed\} \text\{geodesic\} \text\{in\}~\partial M~\text\{of\} \text\{length\}~ \pm L\Big \rbrace \end\{\}\begin\{\}\cup \Big \lbrace \{n\pi \over \omega \}:\text\{there\} \text\{exists\} \text\{a\} \text\{geodesic\}~n\text\{-gon\} \text\{in\}~M~\text\{with\} \text\{vertices\}~\partial M\Big \rbrace \cup \lbrace 0\rbrace .\end\{\}Let $U(t)=\{\rm e\}^\{it(\{1\over 2\}\Delta +V)\}$ be the solution operator to the Cauchy problem for (0.1). Then under a non-trapping assumption for the geodesic flow on $\{\mathrel \{\mathop \{\hspace\{0.0pt\}M\}\limits ^\{\circ \}\}\}$, we have\begin\{\}\{\rm supp~sing~Tr\}\,U(t)\subset S\_\omega ,\end\{\}where $\{\rm Tr\}\,U(t)$ is the distribution given by integrating the Schwartz kernel of $U(t)$ over the diagonal in $M\times M$ or, alternatively, by $\sum _j\{\rm e\}^\{-it\lambda _j\}$, where $\lambda _j$ are the eigenvalues of $\{1\over 2\}\Delta +V$.},
author = {Wunsch, Jared},
journal = {Annales de l'institut Fourier},
keywords = {Schrödinger equation; harmonic oscillator; propagation of singularities; scattering metric; trace theorem},
language = {eng},
number = {1},
pages = {351-373},
publisher = {Association des Annales de l'Institut Fourier},
title = {The trace of the generalized harmonic oscillator},
url = {http://eudml.org/doc/75340},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Wunsch, Jared
TI - The trace of the generalized harmonic oscillator
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 1
SP - 351
EP - 373
AB - We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator\begin{}\Big (D_t+{1\over 2}\Delta +V\Big )\psi =0\qquad (0.1)\end{}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 ${\Bbb R}^n$, radially compactified to the ball) and $V$ is a perturbation of ${1\over 2}\omega ^2x^{-2}$, 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\begin{}S_\omega =\Big \lbrace {L\over \omega }:\text{there} \text{exists} \text{a} \text{closed} \text{geodesic} \text{in}~\partial M~\text{of} \text{length}~ \pm L\Big \rbrace \end{}\begin{}\cup \Big \lbrace {n\pi \over \omega }:\text{there} \text{exists} \text{a} \text{geodesic}~n\text{-gon} \text{in}~M~\text{with} \text{vertices}~\partial M\Big \rbrace \cup \lbrace 0\rbrace .\end{}Let $U(t)={\rm e}^{it({1\over 2}\Delta +V)}$ be the solution operator to the Cauchy problem for (0.1). Then under a non-trapping assumption for the geodesic flow on ${\mathrel {\mathop {\hspace{0.0pt}M}\limits ^{\circ }}}$, we have\begin{}{\rm supp~sing~Tr}\,U(t)\subset S_\omega ,\end{}where ${\rm Tr}\,U(t)$ is the distribution given by integrating the Schwartz kernel of $U(t)$ over the diagonal in $M\times M$ or, alternatively, by $\sum _j{\rm e}^{-it\lambda _j}$, where $\lambda _j$ are the eigenvalues of ${1\over 2}\Delta +V$.
LA - eng
KW - Schrödinger equation; harmonic oscillator; propagation of singularities; scattering metric; trace theorem
UR - http://eudml.org/doc/75340
ER -

References

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