Functional determinant: Difference between revisions

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Let ''S'' be an elliptic [[differential operator]] with smooth coefficients which is positive on functions of compact support. That is, there exists a constant ''c'' > 0 such that

:<math>\langle\phi,S\phi\rangle \ge c\langle\phi,\phi\rangle</math>

for all compactly supported smooth functions &phi;φ. Then ''S'' has a self-adjoint extension to an operator on ''L''² with lower bound ''c''. The eigenvalues of ''S'' can be arranged in a sequence

:<math>0<\lambda_1\le\lambda_2\le\cdots,\qquad\lambda_n\to\infty.</math>

Then the zeta function of ''S'' is defined by the series:<ref>See {{harvtxt|Osgood|Phillips|Sarnak|1988}}. For a more general definition in terms of the spectral function, see {{harvtxt|Hörmander|1968}} or {{harvtxt|Shubin|1987}}.</ref>

:<math>\zeta_S(s) = \sum_{n=1}^\infty \frac{1}{\lambda_n^s}.</math>

It is known that &zeta;ζ_''S'' has a meromorphic extension to the entire plane.<ref>For the case of the generalized Laplacian, as well as regularity at zero, see {{harvtxt|Berline|Getzler|Vergne|2004|loc=Proposition 9.35}}. For the general case of an elliptic pseudodifferential operator, see {{harvtxt|Seeley|1967}}.</ref> Moreover, although one can define the zeta function in more general situations, the zeta function of an elliptic differential operator (or pseudodifferential operator) is [[Mathematical_jargon#regular|regular]] at <math>s = 0</math>.

Formally, differentiating this series term-by-term gives

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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 <math> \sum_{n=0}^{\infty} \frac{1}{(n+a)} </math> , integration over 'a' gives <math> \sum_{n=0}^{\infty}log(n+a) </math> which it just can be considered as the logarithm of the determinant for an [[Harmonic oscillator]] this 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==

[[Image:Infinite potential well.svg|thumb|The infinte potential well with ''A'' = 0.]]

===The infinite potential well===

We will compute the determinant of the following operator describing the motion of a [[quantum mechanics|quantum mechanical]] particle in an [[particle in a box|infinite potential well]]:

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<references/>

*{{Citation | last1=Berline | first1=Nicole | last2=Getzler | first2=Ezra | last3=Vergne | first3=Michèle | title=Heat Kernels and Dirac Operators | isbn=978-3-540-20062-8 | year=2004}}

* {{Citation | last1=Branson | first1=Thomas P. | title=Q-curvature, spectral invariants, and representation theory | idmr={{MathSciNet | id = 2366932}} | year=2007 | journal=SIGMA. Symmetry, Integrability and Geometry. Methods and Applications | issn=1815-0659 | volume=3 | pages=Paper 090, 31}}

* {{Citation | last1=Branson | first1=Thomas P. | title=The functional determinant | publisher=Seoul National University Research Institute of Mathematics Global Analysis Research Center | ___location=Seoul | series=Lecture Notes Series | idmr={{MathSciNet | id = 1325463}} | year=1993 | volume=4}}

* {{Citation | last1=Hörmander | first1=Lars | author1-link=Lars Hörmander | title=The spectral function of an elliptic operator | idmr={{MathSciNet | id = 0609014}} | year=1968 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=121 | pages=193–218 | doi=10.1007/BF02391913}}

* {{Citation | last1=Osgood | first1=B. | last2=Phillips | first2=R. | last3=Sarnak | first3=Peter | authorlink3=Peter Sarnak| title=Extremals of determinants of Laplacians | idmr={{MathSciNet | id = 960228}} | year=1988 | journal=Journal of Functional Analysis | issn=0022-1236 | volume=80 | issue=1 | pages=148–211 | doi=10.1016/0022-1236(88)90070-5}}

* {{Citation | last1=Ray | first1=D. B. | last2=Singer | first2=I. M. |authorlink2=Isadore Singer| title=''R''-torsion and the Laplacian on Riemannian manifolds. | doi=10.1016/0001-8708(71)90045-4 | idmr={{MR|0295381}} | year=1971 | journal=Advances in Math. | volume=7 | pages=145–210}}

* {{Citation | last1=Seeley | first1=R. T. | title=Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) | publisher=[[American Mathematical Society]] | ___location=Providence, R.I. | idmr={{MathSciNet | id = 0237943}} | year=1967 | chapter=Complex powers of an elliptic operator | pages=288–307}}

*{{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 | idmr={{MathSciNet | id = 883081}} | year=1987}}

[[Category:Determinants]]