The ubiquitous $ζ$-function and some of its "usual" and "unusual" meromorphic properties

Exotic expansions and pathological properties of $ζ$-functions on conic manifolds

We give a complete classification and present new exotic phenomena of the meromorphic structure of $ζ$-functions associated to general self-adjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these $ζ$-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. The corresponding heat kernel and resolvent trace expansions also exhibit exotic behaviors with logarithmic terms of arbitrary positive and negative multiplicity. We also give …

具有Pöschl--Teller势的Schrödinger算子的谱$\zeta$-函数的奇特结构

The exotic structure of the spectral $ζ$-function for the Schrödinger operator with Pöschl--Teller potential

本研究通过围道积分构造Pöschl--Teller势Schrödinger算子的谱ζ函数，揭示其具有对数分支点等奇特结构，并表明势的扰动会显著影响该函数的亚纯性质。