nLab Fredholm determinant

Idea

The notion of Fredholm determinants is a generalization of that of determinants of a finite-dimensional matrices to a class of operators on Banach spaces which differ from identity by a trace class operator (or by an appropriate analogue in more abstract context, there are appropriate determinants on certain Banach ideals).

This is often considered as an analytic function of a perturbation parameter $\lambda$.

The calculations with Fredholm determinants have applications in operator theory, random matrix theory, integrable models and elsewhere.

Definition

In his original setup, Fredholm attached the determinant

$$d(\lambda) = \sum_{n=0}^{\infty} \frac{\lambda^n}{n!}\int_a^b\int_a^b\ldots \int_a^b det((K(x_k, x_l))_{k,l=1}^n) d x_1 \cdots d x_n$$

to the Fredholm integral equation of the second kind

$$u(x) + \lambda \int^b_a K(x,y) u(y) d y = f(x), \,\,\,x\in (a,b),$$

and $d$ is an entire function of $\lambda$ such that $d(\lambda) \neq 0$ iff the integral equation has a unique solution.

References

Named after

• Ivar Fredholm: Sur une classe d’équations fonctionnelles, Acta Math. 27 (1903) [doi:10.1007/BF02421317, euclid:10.1007/BF02421317]

General and mathematical

• Wikipedia: Fredholm determinant

• Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library 20; Eigenvalues and s-Numbers, Cambridge Studies in Advanced Mathematics 13; History of Banach spaces and linear operators, sec 6.5.2

• Alexander Grothendieck, La théorie de Fredholm, Bulletin de la Société Mathématique de France 84 (1956), p. 319-384, numdam

• Fritz Gesztesy, Yuri Latushkin, Konstantin A. Makarov, Evans functions, Jost functions and Fredholm determinants, Arch. Rational Mech. Anal. 186 (2007) 361–421,doi pdf

• Barry Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math. 24 (1977), no. 3, 244–273 MR482328 <a href=“http://dx.doi.org/10.1016/0001-8708(77)90057-3”>doi</doi>

• Folkmar Bornemann, On the numerical evaluation of Fredholm determinants, Math. Comp. 79 (2010), no. 270, 871–915 MR2600548 pdf doi

Physics applications

• F. J. Dyson, Fredholm determinants and inverse scattering problems, Comm. Math. Phys. 47, 171–183 (1976) MR406201 euclid

• C. A. Tracy, H. Widom, Fredholm determinants, differential equations and matrix models, Commun. Math. Phys. 163, 33–72 (1994) MR1277933 euclid

• J. Harnad, Alexander R. Its, Integrable Fredholm operators and dual isomonodromic deformations, Comm. Math. Phys. 226 (2002), no. 3, 497–530 MR1896879 doi

• M. Adler, M. Cafasso, P. van Moerbeke, Nonlinear PDEs for Fredholm determinants arising from string equations, arxiv/1207.6341

• M. Bertola, M. Cafasso, Fredholm determinants and pole-free solutions to the noncommutative Painlevé II equation, Comm. Math. Phys. 309 (2012), no. 3, 793–833 MR2885610 doi

• Michio Jimbo, Tetsuji Miwa, Yasuko Môri, Mikio Sato,

  Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1980), no. 1, 80–158 MR573370

category: functional analysis