We explore the meromorphic structure of the $ζ$-function associated to the boundary eigenvalue problem of a modified Sturm-Liouville operator subject to spectral dependent boundary conditions at one end of a segment of length $l$. We find that it presents isolated simple poles which follow the general rule valid for second order differential operators subject to standard local boundary conditions. We employ our results to evaluate the determinant of the operator and the Casimir energy of the system it describes, and study its dependence on $l$ for both the massive and the massless cases.
Boundary Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-Dependent Bulk and Boundary Coupling Strengths
Condensed Matter Theory Center, Department of Physics, University of Maryland（马里兰大学物理系凝聚态理论中心）；Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理与天文学系马尼·L·巴乌米克理论物理研究所）
Conformal Bootstrap for surfaces with boundary in Liouville CFT. Part II: spectral resolution and bootstrap
Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学）；Aix-Marseille University, CNRS, Institut de Mathématiques de Marseille (I2M), and Institut Universitaire de France (IUF)（艾克斯-马赛大学）；Beijing Institute of Technology（北京理工大学）
Boundary conditions effects are explored for size-dependent models in thermal equilibrium. Scalar and fermionic models are used for $D=1+3$ (films), $D=1+2$ (hollow cylinder) and $D=1+1$ (ring). For all models a minimal length is found, below which no thermally-induced phase transition occurs. Using quasiperiodic boundary condition controlled by a contour parameter $θ$ ($θ=0$ is a periodic boundary condition and $θ=1$ is an antiperiodic condition) it results that the minimal length depends directly on the value of $θ$. It is also argued that this parameter can be associated to an Aharonov-Bohm…
