Zeta Determinants on Manifolds with Boundary

Scott, Simon

doi:10.1006/jfan.2001.3893

Cited by 19 publications

(12 citation statements)

References 35 publications

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“…Note 34 One may break the product in (18) into sporadic terms together with terms associated with z n which are close to one of the straight lines already described. The products associated with each line are asymptotically similar to certain formulae involving gamma functions, namely e (γ+δ)z Γ (a + 1)…”

Section: Inverse Spectral Theorymentioning

confidence: 85%

“…The statement of the theorem is a consequence of the fact that F (z) is an entire function of order 1, and can therefore be written in the form F (z) = z m e hz ∞ n=1 1 − z z n e z/zn (18) for some h ∈ C, where z n are the zeros of F (z). See [12, p. 199].…”

Section: Inverse Spectral Theorymentioning

confidence: 99%

“…Problems of this type arise in several contexts. One relates to nonequilibrium thermodynamics, [20] and another concerns the study of generalized determinants for elliptic operators on manifolds with boundary, [13,17,18]. Of course problems involving higher order differential operators can also be reformulated in terms of first order systems; we do not, however, assume Hamiltonian structure, which would force n to be even and also imply strong constraints on the solutions of the equations.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalues of an elliptic system

Davies¹

2003

Mathematische Zeitschrift

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We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained will have to be incorporated into any future general theory.

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Section: Inverse Spectral Theorymentioning

confidence: 85%

Section: Inverse Spectral Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Eigenvalues of an elliptic system

Davies¹

2003

Mathematische Zeitschrift

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“…For generalized APS spectral projectors, the comparison problem for the eta invariant was first solved modulo Z by Lesch and Wojciechowski [20] and Müller [31]; later, this integer ambiguity was removed by the first author [21]. For P − ∈ Gr * ∞ (D − ) and assuming the invertibility of D P− , the comparison problems for the eta invariant and the ζ-determinant were solved by Scott and Wojciechowski [38] and Scott [37]; cf. Forman [10].…”

Section: The Comparison or Relative Invariant Problemmentioning

confidence: 99%

On gluing formulas for the spectral invariants of Dirac type operators

Loya

Park

2005

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. In this note, we announce gluing and comparison formulas for the spectral invariants of Dirac type operators on compact manifolds and manifolds with cylindrical ends. We also explain the central ideas in their proofs. The gluing problem for the spectral invariantsSince their inception, the eta invariant and the ζ-determinant of Dirac type operators have influenced mathematics and physics in innumerable ways. Especially with the development of quantum field theory, the behavior of these spectral invariants under gluing of the underlying manifold has become an increasingly important topic. However, the gluing formula for the ζ-determinant of a Dirac Laplacian has remained an open question due to the nonlocal nature of this invariant. In fact, Bleecker and Booss-Bavnbek stated that [2, p. 89] "no precise pasting formulas are obtained but only adiabatic ones." In [23, 24], we give precise gluing formulas for ζ-determinants of Dirac type operators on compact manifolds and manifolds with cylindrical ends, respectively, and moreover we present new and unified derivations of the gluing formulas for both invariants. The purpose of this note is to announce these gluing formulas for the spectral invariants and to indicate the main ideas in their proofs. We also announce a relative invariant formula proved in [25, 24].We begin with describing the gluing problem for compact manifolds.

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“…In the case that ker Ᏸ T = ker Ᏸ −σ = 0, the second formula in Theorem 1.2 can be derived from [Scott 2002;Scott and Wojciechowski 2000]. We emphasize that the term (det ᏸ −σ ) 2 /(det ᏸ T ) 2 in this formula is new, and this factor is nontrivial in general.…”

Section: Introductionmentioning

confidence: 98%