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The Periodic Schrödinger Operators with Potentials in the Morrey Class

✍ Scribed by Zhongwei Shen

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
260 KB
Volume
193
Category
Article
ISSN
0022-1236

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✦ Synopsis

We consider the periodic Schro¨dinger operator ÀD þ VðxÞ in R d ; d53 with potential V in the Morrey class. Let O be a periodic cell for V: We show that, for p 2 ððd À 1Þ=2; d=2; there exists a positive constant e depending only on the shape of O; p and d such that, if lim sup r!0 sup x2O r 2 1 jBðx; rÞj Z Bðx;rÞ jV ðyÞj p dy 1=p oe;

then the spectrum of ÀD þ V is purely absolutely continuous. We obtain this result as a consequence of certain weighted L 2 Sobolev inequalities on the d-torus. It improves an early result by the author for potentials in L d=2 or weak-L d=2 space.

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