- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:14
Journal of Spectral Theory
J. Spectr. Theory
JST
1664-039X
1664-0403
Quantum theory
10.4171/JST
http://www.ems-ph.org/doi/10.4171/JST
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
4
2014
2
On the distribution of perturbations of propagated Schrödinger eigenfunctions
Yaiza
Canzani
McGill University, MONTREAL, CANADA
Dmitry
Jakobson
McGill University, MONTREAL, CANADA
John
Toth
McGill University, MONTREAL, CANADA
Eigenfunctions, Schrödinger operators, Loschmidt echo, randomwave conjecture, conformal and volume-preserving deformations
Let $(M,g_0)$ be a compact Riemmanian manifold of dimension $n$. Let $P_0 (\h) := -\h^2\Delta_{g}+V$ be the semiclassical Schr\"{o}dinger operator for $\h \in (0,\h_0]$, and let $E$ be a regular value of its principal symbol. % $p_0(x,\xi)=|\xi|^2_{g_0(x)} +V(x)$. Write $\varphi_\h$ for an $L^2$-normalized eigenfunction of $P_0(\h)$ with eigenvalue $E(\h) \in [E-o(1),E+ o(1)]$. Consider a smooth family of metric perturbations $g_u$ of $g_0$ with $u$ in the $k$-ball $ B^k(\varepsilon) \subset \mathbb R^k$ of radius $\varepsilon>0$. For $P_{u}(\h) := -\h^2 \Delta_{g_u} +V$ and small $|t|>0$, we define the propagated perturbed eigenfunctions $$\varphi_\h^{(u)}:=e^{-\frac{i}{\h}t P_u(\h) } \varphi_\h.$$ They appear in the mathematical description of the Loschmidt echo effect in physics. Motivated by random wave conjectures in quantum chaos, we study the distribution of the real part of the perturbed eigenfunctions regarded as random variables $\Re (\varphi^{(\cdot)}_\h(x)): B^{k}(\varepsilon) \to \mathbb R$ for $x\in M$. In particular, when $(M,g)$ is chaotic, we compute the $h \to 0^+$ asymptotics of the variance $\text{Var} [\Re(\varphi^{(\cdot)}_\h(x))] $ and show that the odd moments vanish as $h \to 0^+.$
Partial differential equations
Global analysis, analysis on manifolds
Quantum theory
283
307
10.4171/JST/70
http://www.ems-ph.org/doi/10.4171/JST/70