On the Determinant of One-Dimensional Elliptic Boundary Value Problems

“…[19,23,52,53,54,55,71,73], as well as in the context of the Reidemeister-Franz torsion [97,98]. In particular, in one dimension rather general and elegant results may be obtained, which has attracted the interest of mathematicians especially in the last decade or so [21,22,51,60,61,82,83,84]. In higher dimensions known results are restricted to highly symmetric configurations [13,14,16,23,44,45,46,50] or conformally related ones [10,11,16,47,48].…”

Section: Introductionmentioning

confidence: 99%

Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

2007

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Abstract. In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For arbitrary self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with [13], a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.

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“…Unfortunately, Gelfand and Yaglom's method becomes rather complicated for the periodic and antiperiodic boundary conditions of quantum statistics (see Section 2.12 in [1]), and has therefore rarely been used. Several papers have studied the functional determinants of second-order Sturm-Liouville operators with periodic boundary conditions [3]- [6], and related them to boundary-value problems. The calculated determinants are all singular and were regularized with the help of generalized zeta-functions [7].…”

Section: Introductionmentioning

confidence: 99%

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

Kleinert

Chervyakov

1998

Physics Letters A

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Quadratic fluctuations require an evaluation of ratios of functional determinants of second-order differential operators. We relate these ratios to the Green functions of the operators for Dirichlet, periodic and antiperiodic boundary conditions on a line segment. This permits us to take advantage of Wronski's construction method for Green functions without knowledge of eigenvalues. Our final formula expresses the ratios of functional determinants in terms of an ordinary 2 × 2 -determinant of a constant matrix constructed from two linearly independent solutions of a the homogeneous differential equations associated with the second-order differential operators. For ratios of determinants encountered in semiclassical fluctuations around a classical solution, the result can further be expressed in terms of this classical solution.In the presence of a zero mode, our method allows for a simple universal regularization of the functional determinants. For Dirichlet's boundary condition, our result is equivalent to Gelfand-Yaglom's.Explicit formulas are given for a harmonic oscillator with an arbitrary time-dependent frequency.

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On the Determinant of One-Dimensional Elliptic Boundary Value Problems

Abstract: We discuss the $\zeta-$regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.Comment: LaTeX, 18 page

Cited by 36 publications

References 12 publications

Zeta Determinants on Manifolds with Boundary

Zeta Determinants on Manifolds with Boundary

Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

Simple explicit formulas for Gaussian path integrals with time-dependent frequencies

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