DECAY ESTIMATES FOR FOUR DIMENSIONAL SCHRODINGER, KLEIN-GORDON AND WAVE EQUATIONS WITH OBSTRUCTIONS AT ZERO ENERGY

作者：Green William R^*; Toprak Ebru

来源：Differential and Integral Equations, 2017, 30(5-6): 329-386.

We investigate dispersive estimates for the Schrodinger operator H = -Delta + V with V is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion
e(itH)chi(H)P-ac(H) = O(1/log t)A(0) + O(1/t)A(1) + O((t log t)(-1))A(2) + O(t(-1) (log t)(-2))A(3).
Here, A(0), A(1) : L-1(R-4) -> L-infinity (R-4), while A(2), A(3) are operators between logarithmically weighted spaces, with A(0), A(1), A(2) finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the vertical bar t vertical bar(-2) bound as an operator from L-1 -> L-infinity. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

• 出版日期2017-6