Quasimode and Strichartz estimates for time-dependent Schrödinger equations with singular potentials

Strichartz inequalities with white noise potential on compact surfaces

“…These results can be thought of as quantifying the statement "finite frequencies travel at finite speeds -in (frequency dependent) short time the evolution is morally on flat space". Let us also mention at this point the recent work by Huang and Sogge [20] which deals with a similar setting, however their notion of singular potential refers to low integrability while in our case singular refers rather to potentials with low regularity.…”

Uniform Sobolev estimates in $\mathbb{R}^{n}$ involving singular potentials

“…Inequalities (3.11) and (3.12) are analogs of those in Theorem 1.2 in [7], which, as we shall see later, can be proved by using a similar argument. Inequality (3.13) is the main new ingredient, which will allow us to deal with forcing terms involving bounded and compactly supported potentials.…”

“…The proof of (1.17) requires more work since we can not bound the term II as before if the support of s is unbouned. To proceed, we shall follow the strategy in a recent work [7] by the authors and prove an analogous dyadic estimates which will allow us to obtain (1.17). Also, we have to show that the Littlewood-Paley estimates for H V are valid for the exponents q as in (1.16).…”

“…Also, we have to show that the Littlewood-Paley estimates for H V are valid for the exponents q as in (1.16). This was done in the appendix of [7] in the setting of compact manifolds and the same argument can be used to handle the Euclidean space R n .…”

“…As in [7], the proof of dyadic variants of (1.16) rely on certain microlocalized "quasimode" estimates for the unperturbed scaled Schröding operators with a damping term, (3.8) iλ∂ t + ∆ + iελ.…”