scholarly journals Resonances for Perturbed Periodic Schrödinger Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Mouez Dimassi
Keyword(s):  
Lower Bound ◽  
Schrödinger Operator ◽  
Positive Constant ◽  
Periodic Potential ◽  
Counting Function ◽  
Schrodinger Operator ◽  
Small Positive Constant ◽  
Semiclassical Regime ◽  
Perturbed Periodic ◽  
Periodic Schrödinger Operator

In the semiclassical regime, we obtain a lower bound for the counting function of resonances corresponding to the perturbed periodic Schrödinger operatorPh=-Δ+Vx+W(hx). HereVis a periodic potential,Wa decreasing perturbation andha small positive constant.

Resonances for Slowly Varying Perturbations of a Periodic Schrödinger Operator

2002 ◽  
Vol 54 (5) ◽  
pp. 998-1037 ◽  
Author(s):  
Mouez Dimassi
Keyword(s):  
Schrödinger Operator ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Periodic Potential ◽  
Schrodinger Operator ◽  
Small Positive Constant ◽  
Spectral Bands ◽  
Periodic Schrödinger Operator ◽  
Shape Resonances

AbstractWe study the resonances of the operator . Here V is a periodic potential, φ a decreasing perturbation and h a small positive constant. We prove the existence of shape resonances near the edges of the spectral bands of , and we give its asymptotic expansions in powers of .

Group C*-Algebras and the Spectrum of a Periodic Schrödinger Operator on a Manifold

1992 ◽  
Vol 44 (1) ◽  
pp. 180-193 ◽  
Author(s):  
Toshikazu Sunada
Keyword(s):  
Riemannian Manifold ◽  
Real Axis ◽  
Schrödinger Operator ◽  
Periodic Potential ◽  
Fundamental Result ◽  
Schrodinger Operator ◽  
The Real ◽  
C Algebras ◽  
Closed Intervals ◽  
Periodic Schrödinger Operator

The spectrum of the Laplacian or more generally of a Schrödinger operator on an open manifold may have possibly a complicated aspect. For example, a Cantor set in the real axis may appear as the spectrum even for an innocent looking potential on a standard Riemannian manifold (see J. Moser [10]). The fundamental result of the spectral theory of periodic Schrödinger operators, however, says that the picture of the spectrum of a Schrödinger operator on ℝn with a periodic potential is simple; indeed the spectrum consists of a series of closed intervals of the real axis without accumulation, separated in general by gaps outside the spectrum (see M. Reed and B. Simon [13] or M. M. Skriganov [15] for instance).

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