Abstract
Let (M,g) be an n-dimensional, compact Riemannian manifold and \({P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}\) be a semiclassical Schrödinger operator with \({\hbar \in (0,\hbar_0]}\) . Let \({E(\hbar) \in [E-o(1),E+o(1)]}\) and \({(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}\) be a family of L 2-normalized eigenfunctions of \({P_0(\hbar)}\) with \({P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}\) . We consider magnetic deformations of \({P_0(\hbar)}\) of the form \({P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}\) , where \({\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}\) . Here, u is a k-dimensional parameter running over \({B^k(\epsilon)}\) (the ball of radius \({\epsilon}\)), and the family of the magnetic potentials \({(w_u)_{u\in B^k(\epsilon)}}\) satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of \({(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}\) , given by \({(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}\) , where
for \({|t_0|\in (0,\epsilon)}\) ; the latter functions are themselves eigenfunctions of the \({\hbar}\) -elliptic operators \({Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}\) with eigenvalue \({E(\hbar)}\) and \({Q_0(\hbar) = P_{0}(\hbar)}\). Our main result, Theorem1.2, states that for \({\epsilon >0 }\) small, there are constants \({C_j=C_j(M,V,\omega,\epsilon) > 0}\) with j = 1,2 such that
, uniformly for \({x \in M}\) and \({\hbar \in (0,h_0]}\) . We also give an application to eigenfunction restriction bounds in Theorem 1.3.
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Communicated by Jens Marklof.
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Eswarathasan, S., Toth, J.A. Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions. Ann. Henri Poincaré 14, 611–637 (2013). https://doi.org/10.1007/s00023-012-0198-4
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DOI: https://doi.org/10.1007/s00023-012-0198-4