In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a matrix. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class 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.
so (I+T)−1-I is trace class if T is. It has a natural metric given by d(X, Y) = ||X - Y||1, where || · ||1 is the trace-class norm.
If A is trace-class, then is also trace-class with
This shows that the definition of the Fredholm determinant given by
- If A is a trace-class operator.
- defines an entire function such that
- The function det(I + A) is continuous on trace-class operators, with
- One can improve this inequality slightly to the following, as noted in Chapter 5 of Simon:
- If A and B are trace-class then
- The function det defines a homomorphism of G into the multiplicative group C* of nonzero complex numbers (since elements of G are invertible).
- If T is in G and X is invertible,
- If A is trace-class, then
Fredholm determinants of commutators
A function F(t) from (a, b) into G is said to be differentiable if F(t) -I is differentiable as a map into the trace-class operators, i.e. if the limit
exists in trace-class norm.
If g(t) is a differentiable function with values in trace-class operators, then so too is exp g(t) and
This result was used by Joel Pincus, William Helton and Roger Howe to prove that if A and B are bounded operators with trace-class commutator AB -BA, then
Szegő limit formula
If f is a smooth function on the circle, let m(f) denote the corresponding multiplication operator on H.
- Pm(f) - m(f)P
Let T(f) be the Toeplitz operator on H2 (S1) defined by
then the additive commutator
is trace-class if f and g are smooth.
Berger and Shaw proved that
If f and g are smooth, then
is in G.
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:
where PN is the projection onto the subspace of H spanned by 1, z, ..., zN and a0 = 0.
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.
Informal presentation for the case of integral operators
The section below provides an informal definition for the Fredholm determinant of I-T when the trace-class operator T is an integral operator given by a kernel K(x,x) . 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. Since the kernel K may be defined for a large variety of Hilbert spaces and Banach spaces, this is a non-trivial exercise.
The Fredholm determinant may be defined as
where K is an integral operator. The trace of the operator T and its alternating powers is given in terms of the kernel K by
and in general
The Fredholm determinant was used by physicist John A. Wheeler (1937, Phys. Rev. 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).
- Simon, Barry (2005), Trace Ideals and Their Applications, Mathematical Surveys and Monographs, 120, American Mathematical Society, ISBN 0-8218-3581-5
- Wheeler, John A. (1937), On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure, Physical Review, 52, p. 1107
- Bornemann, Folkmar (2010), "On the numerical evaluation of Fredholm determinants", Math. Comp., Springer, 79: 871–915, doi:10.1090/s0025-5718-09-02280-7