Dispersive Estimates for Higher Dimensional Schrödinger Operators with Threshold Eigenvalues I: the Odd Dimensional Case

We investigate L(R) → L∞(Rn) dispersive estimates for the Schrödinger operator H = −∆ + V when there is an eigenvalue at zero energy and n ≥ 5 is odd. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator Ft satisfying ‖Ft‖L1→L∞ . |t|2− n 2 for |t| > 1 such that ‖ePac − Ft‖L1→L∞ . |t| 1−n 2 , for |t| > 1. With stronger decay conditions on the potential it is possible to generate an operatorvalued expansion for the evolution, taking the form ePac(H) = |t| n 2 A−2 + |t| n 2 A−1 + |t| n 2 A0, with A−2 and A−1 finite rank operators mapping L (R) to L∞(Rn) while A0 maps weighted L spaces to weighted L∞ spaces. The leading order terms A−2 and A−1 vanish when certain orthogonality conditions between the potential V and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining |t|− n 2 A0 term also exists as a map from L (R) to L∞(Rn), hence ePac(H) satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.