Quasimode, Eigenfunction and Spectral Projection Bounds for Schrödinger Operators on Manifolds with Critically Singular Potentials

Abstract

We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, $H_V=-\Delta _g+V(x)$, on compact Riemannian manifolds (M, g) of dimension $n\ge 2$, which extend the results of the third author (Sogge 1988) corresponding to the case where $V\equiv 0$. We are able to handle critically singular potentials and consequently assume that $V\in L^{\tfrac{n}{2}}(M)$ and/or $V\in {{\mathcal {K}}}(M)$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $V\equiv 0$ that go back to the third author (Sogge 1988) as well as ones which arose in the work of Kenig et al. (1987) in the study of “uniform Sobolev estimates” in ${{\mathbb {R}}}^n$. We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural $L^p\rightarrow L^p$ spectral multiplier theorems under the assumption that $V\in L^{\frac{n}{2}}(M)\cap {{\mathcal {K}}}(M)$. Moreover, we can also obtain natural analogs of the original Strichartz estimates (1977) for solutions of $(\partial _t^2-\Delta +V)u=0$. We also are able to obtain analogous results in ${{\mathbb {R}}}^n$ and state some global problems that seem related to works on absence of embedded eigenvalues for Schrödinger operators in ${{\mathbb {R}}}^n$ (e.g., Ionescu and Jerison 2003; Jerison and Kenig 1985; Kenig and Nadirashvili 2000; Koch and Tataru 2002; Rodnianski and Schlag 2004).