JournalsjstVol. 7, No. 1pp. 33–86

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

  • Michael Goldberg

    University of Cincinnati, USA
  • William R. Green

    Rose-Hulman Institute of Technology, Terre Haute, USA
Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case cover
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Abstract

We investigate L1(Rn)L(Rn)L^1(\mathbb R^n)\to L^\infty(\mathbb R^n) dispersive estimates for the Schrödinger operator H=Δ+VH=-\Delta+V when there is an eigenvalue at zero energy in even dimensions n6n\geq 6. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator FtF_t satisfying FtL1Lt2n2\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac {n}{2}} for t>1|t|>1 such that

eitHPacFtL1Lt1n2, for t>1.\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac {n}{2}},\,\,\,\,\,\text{ for } |t|>1.

With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form

eitHPac(H)=t2n2A2+t1n2A1+tn2A0,e^{itH} P_{ac}(H)=|t|^{2-\frac {n}{2}}A_{-2}+ |t|^{1-\frac {n}{2}} A_{-1}+|t|^{-\frac {n}{2}}A_0,

with A2A_{-2} and A1A_{-1} mapping L1(Rn)L^1(\mathbb R^n) to L(Rn)L^\infty(\mathbb R^n) while A0A_0 maps weighted L1L^1 spaces to weighted LL^\infty spaces. The leading-order terms A2A_{-2} and A1A_{-1} are both finite rank, and vanish when certain orthogonality conditions between the potential VV and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining tn2A0|t|^{-\frac {n}{2}}A_0 term also exists as a map from L1(Rn)L^1(\mathbb R^n) to L(Rn)L^\infty(\mathbb R^n), hence eitHPac(H)e^{itH}P_{ac}(H) satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

Cite this article

Michael Goldberg, William R. Green, Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues. II: The even dimensional case. J. Spectr. Theory 7 (2017), no. 1, pp. 33–86

DOI 10.4171/JST/155