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Register a new userThe factorization method, self-similar potentials and quantum algebras
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Authors
V.P. Spiridonov
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This is a brief review of the Schrodingers factorization method and its relations to supersymmetric quantum mechanics and its nonlinear (parastatistical, etc) modifications, self-similar infinite soliton potentials, quantum algebras, coherent states, Ising chains, discretized random matrices and 2D lattice Coulomb gases.
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We investigate a (1+1)-dimensional nonlinear field theoretic model with the field potential $V(phi)| = |phi|.$ It can be obtained as the universal small amplitude limit in a class of models with potentials which are symmetrically V-shaped at their minima, or as a continuum limit of certain mechanical system with infinite number of degrees of freedom. The model has an interesting scaling symmetry of the on shell type. We find self-similar as well as shock wave solutions of the field equation in that model.
We investigate Hawking evaporation in a recently suggested picture in which black holes are Bose condensates of gravitons at a quantum critical point. There, evaporation of a black hole is due to two intertwined effects. Coherent excitation of a tachyonic breathing mode is responsible for the collapse of the condensate, while incoherent scattering of gravitons leads to Hawking radiation. To explore this, we consider a toy model of a single bosonic degree of freedom with derivative self-interactions. We consider the real-time evolution of a condensate and derive evaporation laws for two possible decay mechanisms in the Schwinger-Keldysh formalism. We show that semiclassical results can be reproduced if the decay is due to an effective two-body process, while the existence of a three-body channel would imply very short lifetimes for the condensate. In either case, we uncover the existence of scaling solutions in which the condensate is at a critical point throughout the collapse. In the case of a two-body decay we moreover discover solutions that exhibit the kind of instability that was recently conjectured to be responsible for fast scrambling.
The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.
Given a graph $E$, an action of a group $G$ on $E$, and a $G$-valued cocycle $phi$ on the edges of $E$, we define a C*-algebra denoted ${cal O}_{G,E}$, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup $S_{G,E}$ built naturally from the triple $(G,E,phi)$. As a tight C*-algebra, ${cal O}_{G,E}$ is also isomorphic to the full C*-algebra of a naturally occurring groupoid ${cal G}_{tight}(S_{G,E})$. We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which ${cal O}_{G,E}$ is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our ${cal O}_{G,E}$, and many of their known properties are shown to follow from our general theory.
We investigate strongly correlated non-Abelian plasmas out of equilibrium. Based on numerical simulations, we establish a self-similar scaling property for the time evolution of spatial Wilson loops that characterizes a universal state of matter far from equilibrium. Most remarkably, it exhibits a generalized area law which holds for sufficiently large ratio of spatial area and fractional power of time. Performing calculations also for the perturbative regime at higher momenta, we are able to characterize the full nonthermal scaling properties of SU(2) and SU(3) symmetric plasmas from short to large distance scales in terms of two independent universal exponents and associated scaling functions.
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