Fredholm determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator.
The function is named after the mathematician Erik Ivar Fredholm.
Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model.
In the finite-dimensional case, the determinant of an operator can be interpreted as the factor by which it scales the (oriented) volume of a parallelepiped.
In finite dimensions, by expanding the definition of determinant as a sum over permutations,
ranges over all subsets of the index set of
This generalizes to infinite-dimensional Hilbert spaces, and bounded trace-class operators, allowing us to define the Fredholm determinant by
To show that the definition makes sense, note that if
th elementary symmetric function of the singular values of
One can improve this inequality slightly to the following, as noted in (Simon 2005, Chapter 5):
The Fredholm determinant is often applied to integral operators.
The trace is well-defined for these kernels, since these are trace-class or nuclear operators.
To see that this is a special case of the previous section's general definition, note that,
A proper definition requires a presentation showing that each of the manipulations are well-defined, convergent, and so on, for the given situation for which the Fredholm determinant is contemplated.
The original (Fredholm 1903) considered the integral equation
Fredholm proved that this equation has a unique solution iff
is differentiable as a map into the trace-class operators, i.e. if the limit
is a differentiable function with values in trace-class operators, then so too is
Israel Gohberg and Mark Krein proved that if
This result was used by Joel Pincus, William Helton and Roger Howe to prove that if
Harold Widom used the result of Pincus-Helton-Howe to prove that
He used this to give a new proof of Gábor Szegő's celebrated limit formula:
Szegő's limit formula was proved in 1951 in response to a question raised by the work Lars Onsager and C. N. Yang on the calculation of the spontaneous magnetization for the Ising model.
The formula of Widom, which leads quite quickly to Szegő's limit formula, is also equivalent to the duality between bosons and fermions in conformal field theory.
A singular version of Szegő's limit formula for functions supported on an arc of the circle was proved by Widom; it has been applied to establish probabilistic results on the eigenvalue distribution of random unitary matrices.
The Fredholm determinant was used by physicist John A. Wheeler (1937, Phys.
52:1107) to help provide mathematical description of the wavefunction for a composite nucleus composed of antisymmetrized combination of partial wavefunctions by the method of Resonating Group Structure.
This method corresponds to the various possible ways of distributing the energy of neutrons and protons into fundamental boson and fermion nucleon cluster groups or building blocks such as the alpha-particle, helium-3, deuterium, triton, di-neutron, etc.
When applied to the method of Resonating Group Structure for beta and alpha stable isotopes, use of the Fredholm determinant: (1) determines the energy values of the composite system, and (2) determines scattering and disintegration cross sections.
The method of Resonating Group Structure of Wheeler provides the theoretical bases for all subsequent Nucleon Cluster Models and associated cluster energy dynamics for all light and heavy mass isotopes (see review of Cluster Models in physics in N.D. Cook, 2006).
