Singularity theory study of overdetermination in models for L-H transitions

Symmetry-Breaking Transition and Spectral Singularity in Coupled $\mathcal{PT}$-Symmetric Quantum Potentials

We study the scattering properties of $N$ identical one-dimensional localized $\mathcal{PT}$-symmetric potentials, connected in series as well as in parallel. We derive a general transfer matrix formalism for parallel coupled quantum scatterers, and apply that theory to demonstrate that the spectral singularities and $\mathcal{PT}$-symmetric transitions of single scattering cells may be observed in coupled systems, at the same or distinct values of the critical parameters, depending on the connection modes under which the scattering objects are coupled. We analyse the influences of the connect…

通过Borel变换的浓度下界及算术跃迁附近谱测度的定量奇异性

Lower bounds on concentration through Borel transforms and quantitative singularity of spectral measures near the arithmetic transition

本文开发工具研究双曲区域算术跃迁附近的奇异连续谱，首次给出谱测度维数的跃迁捕捉上界，并通过广义本征函数的部分局域化证明，还提出基于Borel变换边界行为的浓度下界一般准则。

Emergence of a singularity for Toeplitz determinants and Painleve V

We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter $t$. For $t$ positive, the symbols are regular so that the determinants obey Szegő's strong limit theorem. If $t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlevé V transcendent. A particular case of our result complements the classical description of…

Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network

We consider the ferromagnetic Ising model on a highly inhomogeneous network created by a growth process. We find that the phase transition in this system is characterised by the Berezinskii--Kosterlitz--Thouless singularity, although critical fluctuations are absent, and the mean-field description is exact. Below this infinite order transition, the magnetization behaves as $exp(-const/\sqrt{T_c-T})$. We show that the critical point separates the phase with the power-law distribution of the linear response to a local field and the phase where this distribution rapidly decreases. We suggest that…

Regularization of the big bang singularity with a time varying equation of state $w > 1$

We study the classical dynamics of the universe undergoing a transition from contraction to expansion through a big bang singularity. The dynamics is described by a system of differential equations for a set of physical quantities, such as the scale factor $a$, the Hubble parameter $H$, the equation of state parameter $w$, and the density parameter $Ω$. The solutions of the dynamical system have a singularity at the big bang. We study if the solutions can be regularized at the singularity in the sense of whether they have unique branch extensions through the singularity. In particular, we cons…

The Singularity Threshold of the Nonlinear Sigma Model Using 3D Adaptive Mesh Refinement

Numerical solutions to the nonlinear sigma model (NLSM), a wave map from 3+1 Minkowski space to S^3, are computed in three spatial dimensions (3D) using adaptive mesh refinement (AMR). For initial data with compact support the model is known to have two regimes, one in which regular initial data forms a singularity and another in which the energy is dispersed to infinity. The transition between these regimes has been shown in spherical symmetry to demonstrate threshold behavior similar to that between black hole formation and dispersal in gravitating theories. Here, I generalize the result by …