Math 2.0

Global analysis, analysis on manifolds Definition

Zeta-regularized determinant

Builds on Zeta function regularizationElliptic operator on a compact manifoldSelf-adjoint operatorMinakshisundaram–Pleijel zeta function

Definition

Let MM be a closed smooth manifold of dimension nn with a Hermitian vector bundle EE. Let AA be a formally self-adjoint elliptic differential operator of order m>0m > 0 on EE, with nonnegative spectrum. Let 0<μ1≤μ2≤⋯0 < \mu_1 \le \mu_2 \le \cdots be its nonzero eigenvalues, counted with multiplicity. Its spectral zeta function ζA(s)=∑j≥1μj−s\zeta_A(s) = \sum_{j \ge 1} \mu_j^{-s} converges for Re⁡s>n/m\operatorname{Re} s > n/m and has a meromorphic continuation to ℂ\C that is holomorphic at s=0s = 0. The zeta-regularized determinant of AA is det⁡ζ′A=exp⁡(−ζA′(0)).\det{}'_\zeta A = \exp\big( -\zeta_A'(0) \big). If ker⁡A=0\ker A = 0 it is written det⁡ζA\det_\zeta A.

Remarks

It is the zeta-regularized product of the nonzero eigenvalues. For the Laplace–Beltrami operator, ζA\zeta_A is the Minakshisundaram–Pleijel zeta function. If AA were a positive definite matrix, −ζA′(0)=∑jlog⁡μj-\zeta_A'(0) = \sum_j \log \mu_j and the definition would give the usual determinant. The regularized determinant is not multiplicative in general: det⁡ζ(AB)\det_\zeta(AB) may differ from det⁡ζAdet⁡ζB\det_\zeta A \det_\zeta B by a multiplicative anomaly.

Example: on S1=ℝ/2πℤS^1 = \R / 2\pi\Z let A=−d2/dθ2A = -d^2/d\theta^2. The nonzero eigenvalues are k2k^2 with multiplicity 22, so ζA(s)=2ζ(2s)\zeta_A(s) = 2\zeta(2s) and ζA′(0)=4ζ′(0)=−2log⁡(2π)\zeta_A'(0) = 4\zeta'(0) = -2 \log(2\pi), using ζ′(0)=−12log⁡(2π)\zeta'(0) = -\tfrac12 \log(2\pi). Hence det⁡ζ′A=4π2\det{}'_\zeta A = 4\pi^2.

Non-example: the operator −d2/dθ2−c-d^2/d\theta^2 - c with 0<c<10 < c < 1 on S1S^1 has the negative eigenvalue −c-c. It does not fit the definition, since μ−s\mu^{-s} for μ<0\mu < 0 needs a choice of branch of the logarithm.

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Elliptic operator on a compact manifold, concept, no evidence levelElliptic operator on …conceptMinakshisundaram–Pleijel zeta function, concept, no evidence levelMinakshisundaram–Plei…conceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptZeta function regularization, concept, no evidence levelZeta function regular…conceptZeta-regularized determinant, definition, no evidence levelZeta-regularized dete…definition
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  1. Elliptic operator on a compact manifold concept
  2. Minakshisundaram–Pleijel zeta function concept
  3. Self-adjoint operator concept
  4. Zeta function regularization concept
  5. Zeta-regularized determinant definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Functional_determinant
About this page
TagGTJ6KBD4PX
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Kinddefinition
WritingW0 stub
MSC58J52, 11M41
DefinesZeta-regularized determinant
UsesZeta function regularization, Elliptic operator on a compact manifold, Self-adjoint operator, Minakshisundaram–Pleijel zeta function

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