Unusual poles of the $ζ$-functions for some regular singular differential operators

Random matrix ensembles with singularities and a hierarchy of Painlevé III equations

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order $k$ at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierar…

Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure \begin{equation*} \frac{1}{C_n} e^{- n\textrm{tr}\, V(M;λ,\vec{t}\;)}dM, \end{equation*} over the space of $n \times n$ Hermitian matrices $M$, where $V(x;λ,\vec{t}\;):= 2x^2 + \sum_{k=1}^{2m}t_k(x-λ)^{-k}$ with $\vec{t}= (t_1, t_2, \ldots, t_{2m})\in \mathbb{R}^{2m-1} \times (0,\infty)$, in the multiple scaling limit where $λ\to 1$ together with $\vec{t} \to \vec{0}$ as $n\to \infty$ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly …

Resonance index and singular spectral shift function

This paper is a continuation of my previous work on absolutely continuous and singular spectral shift functions, where it was in particular proved that the singular part of the spectral shift function is an a.e. integer-valued function. It was also shown that the singular spectral shift function is a locally constant function of the coupling constant $r,$ with possible jumps only at resonance points. Main result of this paper asserts that the jump of the singular spectral shift function at a resonance point is equal to the so-called resonance index, --- a new (to the best of my knowledge) noti…

On the Heisenberg condition in the presence of redundant poles of the S-matrix

For the same potential as originally studied by Ma [Phys. Rev. {\bf 71}, 195 (1947)] we obtain analytic expressions for the Jost functions and the residui of the S-matrix of both (i) redundant poles and (ii) the poles corresponding to true bound states. This enables us to demonstrate that the Heisenberg condition is valid in spite of the presence of redundant poles and singular behaviour of the S-matrix for $k\to \infty$. In addition, we analytically determine the overall contribution of redundant poles to the asymptotic completeness relation, provided that the residuum theorem can be applied.…

A new look at the Helmholtz equation: Lefschetz thimbles and the einbein action

Picard-Lefschetz theory is applied to solutions of the Helmholtz equation, formulated in terms of sums of integrals of a proper-time, or `einbein', wave function $Ψ(Λ) = \exp(i\mathbb S(Λ))$ along complex contours bounded by essential singularities of $Ψ$. There is a one to one map between steepest descent paths connecting essential singularities and real or complex eigenrays. Residues of finite $Λ$ poles of $\mathbb S(Λ)$ are shown to vanish at spatial points corresponding to sources, provided that the pole bounds only one steepest descent path. If the sum includes two such paths, with one be…

Exceptional points for associated Legendre functions of the second kind

We consider the complex $ν$ plane structure of the associated Legendre function of the second kind $Q^{-1/2-K}_ν(\coshρ)$. We find that for any noninteger value for $K$ $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of poles in the complex $ν$ plane, but for any negative integer $K$ there are no poles at all. For $K=0$ or any positive integer $K$ there is only a finite number of poles, with there only being one single pole (at $ν=0$) when $K=0$. This pattern is characteristic of the exceptional points that appear in a wide variety of physical contexts. However, unusually for theories with excep…

奇异 Liouville 方程非简单爆破解的高阶消失定理

High Order Vanishing Theorems for Nonsimple Blowup Solutions of Singular Liouville Equation

本文研究奇异 Liouville 方程中非简单爆破现象，证明其仅在系数函数导数趋于零时发生，确定了任意量子化奇异源的消失阶数，为消除非简单爆破情形和研究移动极点爆破解提供了理论工具。

SEMPO——从任意光谱响应中提取复频率平面内的极点、留数与零点

SEMPO -- Retrieving poles, residues and zeros in the complex frequency plane from an arbitrary spectral response

本研究提出SEMPO工具箱，依托奇异性展开法结合柯西方法与自动微分优化，无需先验信息即可从实频光谱响应提取线性系统的复极点、零点与留数，可实现大光谱窗口高精度重构，在材料介电常数、超表面光学响应等场景验证了性能。

经典秩一局部对称空间的量子共振与散射极点

Quantum resonances and scattering poles of classical rank one locally symmetric spaces

本文将负曲率对称空间中散射矩阵极点的已知结论（即交织算子极点或量子共振），利用Bunke和Olbrich的结果推广到具有凸紧基本群的负曲率经典局部对称空间，仅排除对应奇异泊松变换的谱参数。

A singular-potential random matrix model arising in mean-field glassy systems

We consider an invariant random matrix model where the standard Gaussian potential is distorted by an additional single pole of order $m$. We compute the average or macroscopic spectral density in the limit of large matrix size, solving the loop equation with the additional constraint of vanishing trace on average. The density is generally supported on two disconnected intervals lying on the two sides of the pole. In the limit of having no pole, we recover the standard semicircle. Obtained in the planar limit, our results apply to matrices with orthogonal, unitary or symplectic symmetry alike.…