Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in ${ℝ}^3$

Erdoğan, M. Burak, Goldberg, Michael, and Schlag, Wilhelm. "Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb {R}^3$." Journal of the European Mathematical Society 010.2 (2008): 507-531. <http://eudml.org/doc/277744>.

@article{Erdoğan2008,
abstract = {We present a novel approach for bounding the resolvent of \[H=-\Delta +i(A\cdot \nabla +\nabla \cdot A)+V=:-\Delta +L 1\] for large energies. It is shown here that there exist a large integer $m$ and a large number $\lambda _0$ so that relative to the usual weighted $L^2$-norm, \[\Vert (L(-\Delta +(\lambda +i0))^\{-1\})^m\Vert <\frac\{1\}\{2\} 2\] for all $\lambda >\lambda _0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and then sum over all possible combinations of cones. Chains of cones that all point in the same direction lead to a Volterra-type gain of the form $(m!)^\{-\varepsilon \}$ with $\varepsilon >0$ fixed. On the other hand, cones that are not aligned contribute little due to the assumed decay of $\hat\{A\}$. We make no use of micro-local analysis, but instead rely on classical phase space techniques. As a corollary of (2), we show that the time evolution of the operator in $\mathbb \{R\}^3$ satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.},
author = {Erdoğan, M. Burak, Goldberg, Michael, Schlag, Wilhelm},
journal = {Journal of the European Mathematical Society},
keywords = {magnetic Schrödinger operators; Strichartz estimates; smoothing estimates; magnetic Schrödinger oeprators; Strichartz estimates; smoothing estimates},
language = {eng},
number = {2},
pages = {507-531},
publisher = {European Mathematical Society Publishing House},
title = {Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb \{R\}^3$},
url = {http://eudml.org/doc/277744},
volume = {010},
year = {2008},
}

TY - JOUR
AU - Erdoğan, M. Burak
AU - Goldberg, Michael
AU - Schlag, Wilhelm
TI - Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb {R}^3$
JO - Journal of the European Mathematical Society
PY - 2008
PB - European Mathematical Society Publishing House
VL - 010
IS - 2
SP - 507
EP - 531
AB - We present a novel approach for bounding the resolvent of \[H=-\Delta +i(A\cdot \nabla +\nabla \cdot A)+V=:-\Delta +L 1\] for large energies. It is shown here that there exist a large integer $m$ and a large number $\lambda _0$ so that relative to the usual weighted $L^2$-norm, \[\Vert (L(-\Delta +(\lambda +i0))^{-1})^m\Vert <\frac{1}{2} 2\] for all $\lambda >\lambda _0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and then sum over all possible combinations of cones. Chains of cones that all point in the same direction lead to a Volterra-type gain of the form $(m!)^{-\varepsilon }$ with $\varepsilon >0$ fixed. On the other hand, cones that are not aligned contribute little due to the assumed decay of $\hat{A}$. We make no use of micro-local analysis, but instead rely on classical phase space techniques. As a corollary of (2), we show that the time evolution of the operator in $\mathbb {R}^3$ satisfies global Strichartz and smoothing estimates without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.
LA - eng
KW - magnetic Schrödinger operators; Strichartz estimates; smoothing estimates; magnetic Schrödinger oeprators; Strichartz estimates; smoothing estimates
UR - http://eudml.org/doc/277744
ER -