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


Let (M,g0)(M,g_0) be a compact Riemmanian manifold of dimension nn. Let P0(\h):=\h2Δg+VP_0 (\h) := -\h^2\Delta_{g}+V be the semiclassical Schr\"{o}dinger operator for \h(0,\h0]\h \in (0,\h_0], and let EE be a regular value of its principal symbol. % p0(x,ξ)=ξg0(x)2+V(x)p_0(x,\xi)=|\xi|^2_{g_0(x)} +V(x). Write φ\h\varphi_\h for an L2L^2-normalized eigenfunction of P0(\h)P_0(\h) with eigenvalue E(\h)[Eo(1),E+o(1)]E(\h) \in [E-o(1),E+ o(1)]. Consider a smooth family of metric perturbations gug_u of g0g_0 with uu in the kk-ball Bk(ε)RkB^k(\varepsilon) \subset \mathbb R^k of radius ε>0\varepsilon>0. For Pu(\h):=\h2Δgu+VP_{u}(\h) := -\h^2 \Delta_{g_u} +V and small t>0|t|>0, we define the propagated perturbed eigenfunctions

φ\h(u):=ei\htPu(\h)φ\h.\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 (φ\h()(x)):Bk(ε)R\Re (\varphi^{(\cdot)}_\h(x)): B^{k}(\varepsilon) \to \mathbb R for xMx\in M. In particular, when (M,g)(M,g) is chaotic, we compute the h0+h \to 0^+ asymptotics of the variance Var[(φ\h()(x))]\text{Var} [\Re(\varphi^{(\cdot)}_\h(x))] and show that the odd moments vanish as h0+.h \to 0^+.

Cite this article

Yaiza Canzani, Dmitry Jakobson, John Toth, On the distribution of perturbations of propagated Schrödinger eigenfunctions. J. Spectr. Theory 4 (2014), no. 2, pp. 283–307

DOI 10.4171/JST/70