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Cubo (Temuco)

versão On-line ISSN 0719-0646

Cubo vol.14 no.1 Temuco  2012

http://dx.doi.org/10.4067/S0719-06462012000100004 

CUBO A Mathematical Journal Vol.14, N° 01, (29-47). March 2012

 

Spectral shift function for slowly varying perturbation of periodic Schrödinger operators.

 

Mouez Dimassi and Maher Zerzeri

Univ. Paris 13, LAGA, (UMR CNRS 7539), F-93430 Villetaneuse, France, email: dimassi@math.univ-paris13.fr

Univ. Paris 13, LAGA, (UMR CNRS 7539), F-93430 Villetaneuse, France, email: zerzeri@math.univ-paris13.fr


ABSTRACT

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schrödinger operators. We give a weak and pointwise asymptotic expansions in powers of h of the derivative of the spectral shift function corresponding to the pair(P(h) = P0 + φ (hx); P0 = -Δ+ v(x)) ; where is a decreasing function, O (|x|-δ) for some δ> n and h is a small positive parameter. Here the potential V is real, smooth and periodic with respect to a lattice T in Rn. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for P(h).

Keywords and Phrases: Periodic Schrödinger operator, spectral shift function, asymptotic expansions, limiting absorption theorem.


RESUMEN

En este artículo estudiamos la expansión asintótica de la función shift espectral para perturbaciones de variación lenta de operadores periódicos de Schrödinger. Proporcionamos una expansión débil y puntual en potencias de h de la derivada de la función shift espectral que corresponde al par (P(h) = P0 + φ (hx); P0 = -Δ+ v(x)) ; donde es una función decreciente, O (|x|-δ) para algún δ > n y h un parámetro positivo pequeño. Aquí el potencial V es real, suave y periódico con respecto a un retículo T in Rn. Para demostrar la expansión asintótica puntual de la función shift espectral establecemos un teorema de absorción límite para P(h).

2010 AMS Mathematics Subject Classification: 81Q10 (35P20 47A55 47N50 81Q15)


 

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Received: February 2011. Revised: March 2011.