Fredholm determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. 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. Definition. Let "H" be a Hilbert  space and "G" the set of bounded invertible operators on "H" of the form "I" + "T", where "T" is a trace-class operator. "G" is a group because formula_1 so ("I"+"T")−"I" is trace class if "T" is. It has a natural metric given by , where is the trace-class norm. If "H" is a Hilbert space with inner product formula_2, then so too is the "k"th exterior power formula_3 with inner product formula_4 In particular formula_5 gives an orthonormal basis of formula_3 if ("e") is an orthonormal basis of "H". If "A" is a bounded operator on "H", then  A functorially defines a bounded operator formula_7 on formula_3 by formula_9 If "A" is trace-class, then formula_7 is also trace-class with formula_11 This shows that the definition of the Fredholm determinant given by formula_12 makes sense. 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 formula_21 exists in trace-class norm. If is a differentiable function with values in trace-class operators, then so too is and formula_22 where formula_23 Israel Gohberg  and Mark Krein proved that if "F" is a differentiable function into "G", then "f" = det "F" is a differentiable map into C* with formula_24 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 formula_25 Szegő limit formula. Let "H" = "L" ("S") and let "P" be the orthogonal projection onto the Hardy space "H" ("S"). If "f" is a smooth function on the circle, let "m"("f") denote the corresponding multiplication operator on "H". The commutator is trace -class. Let "T"("f") be the Toeplitz operator on "H" ("S") defined by formula_26 then the additive commutator formula_27 is trace-class if "f" and "g" are smooth. Berger and Shaw proved that formula_28 If "f" and "g" are smooth, then formula_29 is in "G". Harold Widom used the result of Pincus-Helton-Howe to prove that formula_30 where formula_31 He used this to give a new proof of Gábor Szegő's celebrated limit formula: formula_32 where "P" is the projection onto the subspace of "H" spanned by 1, "z", …, "z" and "a" = 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 formula_33 where "T" is an integral operator. The trace of the  operator "T" and its alternating powers is given in terms of the kernel "K" by formula_34 and formula_35 and in general formula_36 The trace is well-defined for these kernels, since these are trace-class or nuclear operators. Applications. 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).