## Smoothness of the density of states in the Anderson model on a one- dimensional strip.(English)Zbl 0635.60077

Let $$H=-2^{-1}\Delta +V$$ on $$l^ 2({\mathcal D}_ m)$$, where $${\mathcal D}_ m=Z\times \{1,...,m\}$$ and V(x), $$x\in {\mathcal D}_ m$$, are i.i.d. r.v.’s with common probability distribution $$\mu$$. Let $$h(t)=\int e^{-itv}d\mu (v)$$ and let N(E) be the integrated density of states. The folowing are proven.
(i) If h is of class $$C^ d$$ on $$[0,+\infty)$$ with $$4d\geq 3m^ 2-m+4$$ with $$h^{(j)}(t)= O((1+| t|)^{-\alpha})$$ for some $$\alpha >1/2$$ and all $$j=0,1,...,d$$, then N(E) is of class $$C^{[d+1/2]}.$$
(ii) If h is of class $$C^{(1/4)(3m^ 2-m+4)}$$ on $$[0,+\infty)$$ with $$h^{(j)}(t)= O((t+| t|)^{-\alpha})$$ for some $$\alpha >1/2$$ and all $$j=0,1,.., 4^{-1}(3m$$ $$2-m+4)$$, and $$h^ j(t)= O(e^{-b| t|})$$ for $$j=0,1$$ and some $$b>0$$, then $$N(E)$$ has an analytic extension to a strip about the real axis.
The proof uses the supersymmetric replica trick to rewrite the averaged Green’s function as a two-point function of a supersymmetric field theory which is studied by the transfer matrix method.
Reviewer: A.Klein

### MSC:

 60H25 Random operators and equations (aspects of stochastic analysis) 81P20 Stochastic mechanics (including stochastic electrodynamics)
Full Text:

### References:

 [1] Anderson, P., Phys. rev., 109, 1492, (1958) [2] Carmona, R., () [3] Kirsch, W.; Martinelli, F., J. phys. A, 15, 2139, (1982) · Zbl 0492.60055 [4] Benderskii, M.; Pastur, L., Mat. sb., 82, 245, (1970) [5] Avron, J.; Simon, B., Duke math. J., 50, 369, (1983) [6] Spencer, T., () [7] Pastur, L., Commun. math phys., 75, 179, (1980) [8] Delyon, F.; Souillard, B., Commun. math. phys., 94, 289, (1984) [9] Craig, W.; Simon, B., Duke math. J., 50, 551, (1983) [10] Le Page, E., () [11] Carmona, R.; Klein, A.; Martinelli, F., Commun. math. phys., 108, 41, (1987) [12] Simon, B.; Taylor, M., Commun. math. phys., 101, 1, (1985) [13] Campanino, M.; Klein, A., Commun. math. phys., 104, 227, (1984) [14] {\scP. March and A. Sznitman}, Some connections between excursion theory and the discrete random SchrĂ¶dinger equation with applications to analyticity and smoothness properties of the density of states in one dimension, preprint. [15] Edwards, S.; Thouless, D., J. phys. C, 4, 453, (1971) [16] Wegner, F., Z. phys. B, 44, 9, (1981) [17] {\scR. Maier}, J. Stat. Phys., in press. [18] Constantinescu, F.; Frohlich, J.; Spencer, T., J. stat. phys., 34, 571, (1984) [19] {\scA. Bovier, M. Campanino, A. Klein, and F. Perez}, Commun. Math. Phy., in press. [20] Bougerol, P.; Lacroix, J., () [21] {\scA. Klein and A. Speis}, in preparation. [22] Parisi, G.; Sourlas, N., J. phys. lett., 41, L403, (1980) [23] MacKane, A.J., Phys. lett. A, 76, 22, (1980) [24] Luttinger, J.M., J. math. phys., 24, 2070, (1983) [25] Effetov, K.B., Adv. phys., 32, 53, (1983) [26] Klein, A.; Perez, J.F., Nucl. phys. B, 251, FS13, 199, (1985) [27] Klein, A.; Landau, L.J.; Perez, J.F., Commun. math. phys., 93, 459, (1984) [28] Whitney, H., Trans. amer. math. soc., 36, 63, (1934) [29] Stein, E., () [30] Berezin, F.A., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.