Fredholm Theory - Fredholm Determinant

Fredholm Determinant

The Fredholm determinant is commonly defined as

where

and

and so on. The corresponding zeta function is

The zeta function can be thought of as the determinant of the resolvent.

The zeta function plays an important role in studying dynamical systems. Note that this is the same general type of zeta function as the Riemann zeta function; however, in this case, the corresponding kernel is not known. The existence of such a kernel is known as the Hilbert–Pólya conjecture.