PACS. 05.50 -Lattice theory and statistics; Ising problems.Abstract. -Configurational statistics of heteropolymer with quenched disordered sequence of links is investigated with the aid of replica approach. It is shown that frozen states with broken ergodicity exist in this system. The features of nonergodic state as well as the freezing transition depend drastically on the dimensionality of space d. For d > 2 the model is analogous to Potts glass and there exists the first-order phase transition with discontinuous behaviour of the order paxameter-replica overlaps-but without specific heat. The set of states in the frozen phase is not ultrametric and the number of states decreases when the heterogeneity of the chain increases. For d c 2 the transition is of the second order, the set of states is ultrametric and the number of states does not depend on heterogeneity of the polymer.

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Under the assumption that V ∈ L 2 ([0, π]; dx), we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators −d 2 /dx 2 + V in L 2 ([0, π]; dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators −d 2 /dx 2 + V in L p ([0, π]; dx), p ∈ (1, ∞).

We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H = −d 2 /dx 2 + V in L 2 (R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V . To cite this article: F. Gesztesy, V. Tkachenko, C. R. Acad. Sci. Paris, Ser. I 343 (2006). RésuméQuand un opérateur de Hill non-autoadjoint est-il un operateur spectral de type scalaire ? Nous dérivons des conditions nécessaires et suffisantes pour qur l'opérateur de Schrödinger (i.e., l'opérateur de Hill) H = −d 2 /dx 2 + V dans L 2 (R) soit un opérateur spectral de type scalaire. Les conditions montrent que cette propriétés ne dépend pas des propriétés différentielles (ou analytiques) du potentiel V . Pour citer cet article : F. Gesztesy, V. Tkachenko, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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