Functional determinant: Difference between revisions

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:<math>\int_V e^{-\pi\langle \phi,S\phi\rangle}\, \mathcal D\phi</math>

where ''V'' is the function space and <math>\langle \cdot,\cdot\rangle</math> the [[L2 norm|L²]] inner product, and <math>\mathcal D\phi</math> the [[Wiener measure]]. The basic assumption on ''S'' is that it should be self-adjoint, and have discrete [[operator spectrum|spectrum]] λ₁, λ₂, λ₃, …... with a corresponding set of [[eigenfunctions]] ''f''₁, ''f''₂, ''f''₃, …... which are complete in [[Lp space|L²]] (as would, for example, be the case for the second derivative operator on a compact interval Ω). This roughly means all functions φ can be written as [[linear combination]]s of the functions ''f''_''i'':

:<math> |\phi\rangle = \sum_i c_i |f_i\rangle \quad \text{with } c_i = \langle f_i | \phi \rangle. </math>

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Since the analytic continuation of the zeta function is regular at zero, this can be rigorously adopted as a definition of the determinant.

This kind of Zeta-regularized functional determinant also appears when evaluating sums of the form {{nowrap|<math display="inline"> \sum_{n=0}^{\infty} \frac{1}{(n+a)} </math>,}}. integrationIntegration over ''a'' gives <math display="inline"> \sum_{n=0}^{\infty}\ln(n+a) </math> which itcan just can be considered as the logarithm of the determinant for a [[Harmonic oscillator]]. thisThis last value is just equal to <math> -\partial _s \zeta_H(0,a) </math>, where <math> \zeta_H(s,a) </math> is the [[Hurwitz zeta function]].

==Practical example==

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== See also ==

* [[Abstract Wiener space]]

* [[Berezinian]]

* [[Fredholm determinant]]

* [[Fujikawa method]]

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==References==

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*{{Citation | last1=Shubin | first1=M. A. | title=Pseudodifferential operators and spectral theory | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Springer Series in Soviet Mathematics | isbn=978-3-540-13621-7 | mr=883081 | year=1987}}