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

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“…Combining this estimate with as well as Grönwall's inequality, we conclude (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23). Additionally, we arrive at (5-24) if we apply Lemma 5.3 with (5-21) instead of (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) in the above argument.…”

“…. Therefore, combining (5-31) and (5-32) with inequality (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) given in Lemma 5.4 yields, for any t ≤ T ,…”

“…where the constant C ρ > 0 is provided in Proposition 5.5 and the constant C * > 0 depends only on universal constants, ∥ρ in ∥ C α 0 ‫ޔ(‬ d ) , and ∥h ϵ,in ∥ C α 0 ‫ޔ(‬ d ‫ޒ×‬ d ) . Then, using the estimate (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) given in Proposition 5.5 with (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22), we arrive at point (i) of Theorem 1.4.…”

“…In view of Hölder's inequality and inequality (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) given in Lemma 5.4, we know that…”

“…As for point (ii) of Theorem 1.4, applying (5-24) together with (5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) and , for any t ∈ [1, T ], we have…”