Dispersive estimates for higher dimensional Schrodinger operators with threshold eigenvalues II: The even dimensional case

作者：Goldberg Michael^*; Green William R

来源：Journal of Spectral Theory, 2017, 7(1): 33-86.

DOI：10.4171/JST/155

• Schrodinger operator
• eigenvalue
• dispersive estimate
• resolvent

We investigate L-1(R-n) -> L-infinity(R-n) dispersive estimates for the Schrodinger operator H = -Delta+V when there is an eigenvalue at zero energy in even dimensions n >= 6. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator F-t satisfying parallel to F-t parallel to(L1 -> L infinity) less than or similar to vertical bar t vertical bar(2-n/2) for vertical bar t vertical bar > 1 such that
parallel to e(itH) P-ac - F-t parallel to(L1 -> L infinity) less than or similar to vertical bar t vertical bar(1-n/2), for vertical bar t vertical bar > 1.
With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form
e(itH) P-ac(H) = vertical bar t vertical bar(2-n/2)A(-1) + vertical bar t vertical bar(-n/2)A(0),
with A(-2) and A(-1) mapping L-1(R-n) to L-infinity(R-n)while A(0) maps weighted L-1 spaces to weighted L-infinity spaces. The leading-order terms A(-2) and A(-1) are both finite rank, and 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 vertical bar t vertical bar(-n/2) A(0) termalso exists as amap from L-1(R-n) to L-infinity(R-n) hence e(itH) P-ac(H) satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

• 出版日期2017