Destruction of Anderson localization in quantum nonlinear Schr\"odinger lattices

The four-wave interaction in quantum nonlinear Schr\"odinger lattices with disorder is shown to destroy the Anderson localization of waves, giving rise to unlimited spreading of the nonlinear field to large distances. Moreover, the process is not thresholded in the quantum domain, contrary to its "classical" counterpart, and leads to an accelerated spreading of the subdiffusive type, with the dispersion $\langle(\Delta n)^2\rangle \sim t^{1/2}$ for $t\rightarrow+\infty$. The results, presented here, shed new light on the origin of subdiffusion in systems with a broad distribution of relaxation times.Comment: 4 pages, no figure

A topological approximation of the nonlinear Anderson model

We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrodinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance-overlap in phase space, ranging from a fully developed chaos involving Levy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. We describe this localization-delocalization transition as a percolation transition on a Cayley tree. It is found in vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha, with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of stripes propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for publication in Physical Review

Shostakovich, old believers and new minimalists

The chapter discusses ‘minimalist’ elements of Dmitri Shostakovich's style as embodiments/ expressions of traditional Russian expressive modes rooted in the idioms of old folk music and the music of the ‘old believers’ The collection volume comprises a selection of articles that, as a group, marks an important new stage in our understanding of Shostakovich and his working environment. The papers have in common a perspective that we believe offers the most fruitful route forward for Shostakovich studies today. All address aspects of the composer’s output in the context of his life and cultural milieu. They are thus illuminating from two directions: the uncovering of ‘outside’ stimuli allows us to perceive the motivations behind Shostakovich’s artistic choices, while at the same time the nature of those choices offers insights into the workings of the larger world—cultural, social, political—that he inhabited. Thus his often ostensibly quirky choices are revealed as responses—by turns sentimental, moving, sardonic and angry—to the particular conditions, with all their absurdities and contradictions, that he had to negotiate. The composer emerging from the role of tortured loner of older narratives into that of the gregarious and engaged member of his society that, for better and worse, characterized the everyday reality of his life

On the Schatten-von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten-von Neumann classes $S_q$ with $0<q\le 1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.Comment: 22 page