Regularization zeta

The regularization zeta is a method of Régularisation Déterminant S of operator S which appear during integral calculations of of ways in Quantum theory of the fields.

The case of the Laplacian

That is to say $\ Omega$ a compact field of $\ mathbb R^n$ on partial board $\ \ Omega$ . On this field, one considers the positive operator $\ hat {H} = - \ \ Delta$ , where $\ Delta$ is the Laplacien, provided with boundary conditions on the partial edge $\ \ Omega$ of the field (Dirichlet, Neumann, mixed) which specifies the problem completely.

When the field $\ Omega$ is compact, the positive operator $\ hat {H} = - \ \ Delta$ has a discrete Specter of eigenvalues with which a base orthonormée with clean vectors is associated (one uses here the notations of Dirac):

\ hat {H} \ | \ psi_n \ rangle \ = \ \ lambda_n \ | \ psi_n \ rangle \, \ quad 0 \ the \ lambda_1 \ the \ lambda_2 \ the \ dowries \ the \ lambda_n \ the \ dowries \ it + \ infty

Spectral function zeta

Definition

It is supposed here that fundamental the $\ lambda_1 \ 0$ . By analogy with the Function zeta of Riemann, one introduces the spectral function zeta by the series of the type Dirichlet:

\ zeta (S) \ = \ \ sum_ {n=1} ^ {+ \ infty} \ \ frac {1} {\ lambda_n^s}

This series converges only for $\ Re \ mathrm {E} \ left \, S \, \ right$ sufficiently large, but she admits a prolongation Méromorphe in the whole plan. When the spectrum of the operator $\ hat {H}$ is not known explicitly, one can use the formal definition like Trace:

\ zeta (S) \ = \ \ mathrm {Tr} \ \ exp \ \ left \ - \ S \ \ ln \ hat {H} \ \ right

Bond with the determinant

The determinant of the operator H is defined by:

\ mathrm {det} \ \ hat {H} \ = \ \ prod_ {n=1} ^ {+ \ infty} \ \ lambda_n

With the identity:

\ ln \ \ mathrm {det} \ \ hat {H} \ = \ \ ln \ \ left (\ prod_ {n=1} ^ {+ \ infty} \ \ lambda_n \ right) \ = \ \ sum_ {n=1} ^ {+ \ infty} \ \ ln \ lambda_n \ = \ \ mathrm {Tr} \ \ ln \ \ hat {H}

the formal relation easily is shown:

\ mathrm {det} \ \ hat {H} \ = \ \ exp \, \ left \, - \ \ zeta' (0) \, \ right

where the derived from the function zeta is evaluated in S = 0.

Bond with the core of heat

The function zeta is connected by a transformed of type Mellin:

\ zeta (S) \ = \ \ frac {1} {\ Gamma (S)} \ \ int_0^ {+ \ infty} dt \ t^ {s-1} \ \ mathrm {Tr} \ e^ {- \; T \; \ hat {H}}

with the Trace of the Core of heat, defined by:

\ mathrm {Tr} \ e^ {- \; T \; \ hat {H}} \ = \ \ sum_ {n=1} ^ {+ \ infty} \ e^ {- \; T \; \ lambda_n}

Extensions

All the preceding definitions rather naturally transpose to the case of the Opérateur of Laplace-Beltrami on a Variété riemannienne compact E , which then has also a discrete spectrum. They also extend to the case from the not-compact varieties on board when the spectrum is still discrete

It is also possible to extend the theory for another elliptic operator.

Dependant articles

Random links: