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

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Let   be a Hilbert space and   the set of bounded invertible operators on   of the form  , where   is a trace-class operator.   is a group because

 

so   is trace class if   is. It has a natural metric given by  , where   is the trace-class norm.

If   is a Hilbert space with inner product  , then so too is the  th exterior power   with inner product  

In particular  

gives an orthonormal basis of   if   is an orthonormal basis of  . If   is a bounded operator on  , then   functorially defines a bounded operator   on   by  

If   is trace-class, then   is also trace-class with  

This shows that the definition of the Fredholm determinant given by  

makes sense.

Properties

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  • If   is a trace-class operator

  defines an entire function such that  

  • The function   is continuous on trace-class operators, with

  One can improve this inequality slightly to the following, as noted in Chapter 5 of Simon:  

  • If   and   are trace-class then

 

  • The function   defines a homomorphism of   into the multiplicative group   of nonzero complex numbers (since elements of   are invertible).
  • If   is in   and   is invertible,

 

  • If   is trace-class, then

   

Fredholm determinants of commutators

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A function   from   into   is said to be differentiable if   is differentiable as a map into the trace-class operators, i.e. if the limit

 

exists in trace-class norm.

If   is a differentiable function with values in trace-class operators, then so too is   and

 

where  

Israel Gohberg and Mark Krein proved that if   is a differentiable function into  , then   is a differentiable map into   with  

This result was used by Joel Pincus, William Helton and Roger Howe to prove that if   and   are bounded operators with trace-class commutator  , then

 

Szegő limit formula

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Let   and let   be the orthogonal projection onto the Hardy space  .

If   is a smooth function on the circle, let   denote the corresponding multiplication operator on  .

The commutator   is trace-class.

Let   be the Toeplitz operator on   defined by  

then the additive commutator   is trace-class if   and   are smooth.

Berger and Shaw proved that  

If   and   are smooth, then   is in  .

Harold Widom used the result of Pincus-Helton-Howe to prove that   where  

He used this to give a new proof of Gábor Szegő's celebrated limit formula:   where   is the projection onto the subspace of   spanned by   and  .

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

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The section below provides an informal definition for the Fredholm determinant of   when the trace-class operator   is an integral operator given by a kernel  . 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   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   is an integral operator. The trace of the operator   and its alternating powers is given in terms of the kernel   by   and   and in general  

The trace is well-defined for these kernels, since these are trace-class or nuclear operators.

Applications

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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).

References

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  • Simon, Barry (2005), Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, ISBN 0-8218-3581-5
  • Wheeler, John A. (1937-12-01). "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure". Physical Review. 52 (11). American Physical Society (APS): 1107–1122. Bibcode:1937PhRv...52.1107W. doi:10.1103/physrev.52.1107. ISSN 0031-899X.
  • Bornemann, Folkmar (2010), "On the numerical evaluation of Fredholm determinants", Math. Comp., 79 (270), Springer: 871–915, arXiv:0804.2543, doi:10.1090/s0025-5718-09-02280-7