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

M. Burak Erdoğan; Michael Goldberg; Wilhelm Schlag

Journal of the European Mathematical Society (2008)

- Volume: 010, Issue: 2, page 507-531
- ISSN: 1435-9855

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topErdoğan, 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 -

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