scholarly journals Decay rates at infinity for solutions to periodic Schrödinger equations

2019 ◽  
Vol 150 (3) ◽  
pp. 1113-1126
Author(s):  
Daniel M. Elton
Keyword(s):  
Half Space ◽  
Absolute Continuity ◽  
Evolution Equations ◽  
Schrödinger Operators ◽  
Decay Rates ◽  
Schrodinger Equations ◽  
Schrodinger Operators ◽  
Schrödinger Equations ◽  
The Absolute ◽  
Continuity Of The Spectrum

AbstractWe consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT UNBOUNDED SINGULAR POTENTIALS

2011 ◽  
Vol 23 (08) ◽  
pp. 823-838 ◽  
Author(s):  
KENJI YAJIMA
Keyword(s):  
Evolution Equations ◽  
Schrödinger Operators ◽  
Time Variable ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Singular Potentials ◽  
Spatial Infinity ◽  
Abstract Theorem ◽  
Magnetic Potentials

We consider time-dependent perturbations by unbounded potentials of Schrödinger operators with scalar and magnetic potentials which are almost critical for the selfadjointness. We show that the corresponding time-dependent Schrödinger equations generate a unique unitary propagator if perturbations of scalar and magnetic potentials are differentiable with respect to the time variable and they increase at the spatial infinity at most quadratically and at most linearly, respectively, where both have mild local singularities. We use time-dependent gauge transforms and apply Kato's abstract theorem on evolution equations.

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