Jinsung Park - Academia.edu (original) (raw)

Papers by Jinsung Park

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2008

In this paper we announce fundamental results of the Ruelle zeta function for odd dimensional hyp... more In this paper we announce fundamental results of the Ruelle zeta function for odd dimensional hyperbolic manifolds with cusps; the meromorphic extension over C, its functional equation and the singularity at s ¼ 0.

The Michigan Mathematical Journal, 2006

Journal of Physics A: Mathematical and Theoretical, 2008

In this contribution we announce a complete classification and new exotic phenomena of the meromo... more In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of ζ-functions associated to conic manifolds proved in [37]. In particular, we show that the meromorphic extensions of these ζfunctions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

Journal of Physics A: Mathematical and General, 2005

In this paper, we derive a formula for the ratio of the ζ-determinants of the Laplacian with Neum... more In this paper, we derive a formula for the ratio of the ζ-determinants of the Laplacian with Neumann and Dirichlet boundary conditions over a noncompact manifold with an infinite cylindrical end and a compact boundary in terms of the ζ-determinant of the Dirichlet to Neumann map.

Journal of Mathematical Physics, 2006

In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the o... more In this article we analyze the resolvent, the heat kernel and the spectral zeta function of the operator −d 2 /dr 2 −1/(4r 2) over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for k ∈ N we find terms like (log t) −k in the small-t asymptotic expansion of the heat kernel. Furthermore, the zeta function has s = 0 as a logarithmic branch point.

Journal of Mathematical Physics, 2005

In this note, we explicitly compute the functional determinant of a Dirac Laplacian with nonlocal... more In this note, we explicitly compute the functional determinant of a Dirac Laplacian with nonlocal pseudodifferential boundary conditions over a finite cylinder in terms of the ζ-function of the Dirac operator on the cross section and the pseudodifferential operators defining the boundary conditions. In particular, this result reduces to our previous formula [J. Phys. AJPHAC5 37, 7381 (2004)] for the special case of generalized Atiyah–Patodi–Singer conditions. To prove our main result, we use the gluing and comparison formulas established by the present authors in Refs 14 and 15.

Journal of Geometry and Physics, 2005

We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on... more We discuss the adiabatic decomposition formula of the ζ-determinant of a Laplace type operator on a closed manifold. We also analyze the adiabatic behavior of the ζ-determinant of a Dirichlet to Neumann operator. This analysis makes it possible to compare the adiabatic decomposition formula with the Meyer-Vietoris type formula for the ζ-determinant proved by Burghelea, Friedlander and Kappeler. As a byproduct of this comparison, we obtain the exact value of the local constant which appears in their formula for the case of Dirichlet boundary condition.

Journal of Geometry and Physics, 2004

In this note we specialize and illustrate the ideas developed in the paper [Index theory, eta for... more In this note we specialize and illustrate the ideas developed in the paper [Index theory, eta forms, and Deligne cohomology] Families, Eta forms, and Deligne cohomology in the case of the determinant line bundle. We discuss the surgery formula in the adiabatic limit using the adiabatic decomposition formula of the zeta regularized determinant of the Dirac Laplacian in [Scattering theory, the adiabatic decomposition of the ζ-determinant and the Dirichlet to Neumann operator, Preprint].

Journal of Geometric Analysis, 2008

We give a complete classification and present new exotic phenomena of the meromorphic structure o... more We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general selfadjoint 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 a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

Annals of Physics, 2006

In this paper, a complete description of the zeta functions and corresponding zeta determinants f... more In this paper, a complete description of the zeta functions and corresponding zeta determinants for Dirac and Laplace-type operators over finite cylinders using the contour integration method, for example described in [K. Kirsten, Spectral Functions in Mathematics and Physics, Chapman & Hall/ CRC Press, Boca Raton, 2001] is given. Different boundary conditions, local and non-local ones, are considered. The method is shown to be very powerful in that it is easily adapted to each situation and in that answers are very elegantly obtained.

Annals of Global Analysis and Geometry, 2006

The goal of this paper is to establish a geometric program to study elliptic pseudodifferential b... more The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and 'b-admissible' partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderón projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulae, and solve the Bojarski conjecture.

Advances in Mathematics, 2006

In this paper we solve the gluing problem for the ζ-determinant of a Dirac Laplacian. To do so, w... more In this paper we solve the gluing problem for the ζ-determinant of a Dirac Laplacian. To do so, we develop a new approach to solve such problems which relies heavily on the theory of elliptic boundary problems, the analysis of the resolvent of the Dirac operator, and the introduction of an auxiliary model problem. Moreover, as a byproduct of our approach we obtain a new gluing formula for the eta invariant au gratis.

Journal of Physics A: Mathematical and General, 2004

In this note, we explicitly compute the ζ-determinant of a Dirac Laplacian with APS boundary cond... more In this note, we explicitly compute the ζ-determinant of a Dirac Laplacian with APS boundary conditions over a finite cylinder. Using this exact result, we illustrate the gluing and comparison formulas for the ζ-determinants of Dirac Laplacians proved in [12] and [14].