On the distribution of perturbations of propagated Schrödinger eigenfunctions

Abstract

Let $(M,g_0)$ be a compact Riemmanian manifold of dimension $n$. Let $P_0 (\h) := -\h^2Δ_{g}+V$ be the semiclassical Schrödinger operator for $\h \in (0,\h_0]$, and let $E$ be a regular value of its principal symbol $p_0(x,\xi )=|\xi {|}_{g_0(x)}^2+V(x)$. Write $φ_\h$ for an $L^2$-normalized eigenfunction of $P(\h)$, $P_0(\h)φ_\h =E(\h)φ_\h$ and $E(\h) \in [E-o(1),E+ o(1)]$. Consider a smooth family of perturbations $g_u$ of $g_0$ with $u$ in the ball ${\mathcal{B}}^k(\epsilon )\subset {\mathbb{R}}^k$ of radius $\epsilon >0$. For $P_{u}(\h) := -\h^2 Δ_{g_u} +V$ and small $|t|$, we define the propagated perturbed eigenfunctions $$φ_\h^{(u)}:=e^{-\frac{i}{\h}t P_u(\h)} φ_\h.$$ We study the distribution of the real part of the perturbed eigenfunctions regarded as random variables $$\Re (φ^{(\cdot)}_\h(x)):\mathcal B^{k}(\varepsilon) \to \mathbb R \quad \quad \text{for}\;\, x\in M.$$ In particular, when $(M,g)$ is ergodic, we compute the $h\to 0^{+}$ asymptotics of the variance $\text{Var} [\Re (φ^{(\cdot)}_\h(x))] $ and show that all odd moments vanish as $h\to 0^{+}.$

To appear in Journal of Spectral Theory

References in corpus (2)

• Laplacian eigenvalues functionals and metric deformations on compact manifolds
• Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions