Jacobi operator

A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi.

The name derives from a theorem from Jacobi, dating to 1848, stating that every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix.

Self-adjoint Jacobi operators

The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers $\ell^2(\mathbb{N})$ . In this case it is given by

Jf_{0}=a_{0}f_{1}+b_{0}f_{0},\quad Jf_{n}=a_{n}f_{n+1}+b_{n}f_{n}+a_{n-1}f_{n-1},\quad n>0,

where the coefficients are assumed to satisfy

a_{n}>0,\quad b_{n}\in \mathbb {R} .

The operator will be bounded if and only if the coefficients are.

There are close connections with the theory of orthogonal polynomials. In fact, the solution $p_{n}(x)$ of the recurrence relation

J\,p_{n}(x)=x\,p_{n}(x),\qquad p_{0}(x)=1{\text{ and }}p_{-1}(x)=0,

is a polynomial of degree n and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector $\delta _{1,n}$ .

This recurrence relation is also commonly written as

xp_{n}(x)=a_{n+1}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n}p_{n-1}(x)

Applications

It arises in many areas of mathematics and physics. The case a(n) = 1 is known as the discrete one-dimensional Schrödinger operator. It also arises in:

The Lax pair of the Toda lattice.
The three-term recurrence relationship of orthogonal polynomials, orthogonal over a positive and finite Borel measure.
Algorithms devised to calculate Gaussian quadrature rules, derived from systems of orthogonal polynomials.^[1]

Generalizations

When one considers Bergman space, namely the space of square-integrable holomorphic functions over some domain, then, under general circumstances, one can give that space a basis of orthogonal polynomials, the Bergman polynomials. In this case, the analog of the tridiagonal Jacobi operator is a Hessenberg operator – an infinite-dimensional Hessenberg matrix. The system of orthogonal polynomials is given by

zp_{n}(z)=\sum _{k=0}^{n+1}D_{kn}p_{k}(z)

and $p_{0}(z)=1$ . Here, D is the Hessenberg operator that generalizes the tridiagonal Jacobi operator J for this situation.^[2]^[3]^[4] Note that D is the right-shift operator on the Bergman space: that is, it is given by

[Df](z)=zf(z)

The zeros of the Bergman polynomial $p_{n}(z)$ correspond to the eigenvalues of the principle $n\times n$ submatrix of D. That is, The Bergman polynomials are the characteristic polynomials for the principle submatrixes of the shift operator.

References

↑ Fast variants of the Golub and Welsch algorithm for symmetric weight functions – Gérard Meurant
↑ V. Tomeo, E. Torrano, "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials", Linear Algebra and its Applications (2011) Volume 435, Issue 9, Pages 2314-2320, DOI=https://doi.org/10.1016/j.laa.2011.04.027 URL=http://oa.upm.es/id/eprint/8725/contents
↑ Edward B. Saff and Nikos Stylianopoulos, "Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan region", (2012) arXiv:1205.4183 [math.CV] URL=https://arxiv.org/abs/1205.4183
↑ Carmen Escribano and Antonio Giraldo and M. Asunción Sastre and Emilio Torrano, "The Hessenberg matrix and the Riemann mapping" (2011){arXiv:1107.603 [math.SP] URL=https://arxiv.org/abs/1107.6036

Teschl, Gerald (2000), Jacobi Operators and Completely Integrable Nonlinear Lattices, Providence: Amer. Math. Soc., ISBN 0-8218-1940-2

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Jacobi matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Fast variants of the Golub and Welsch algorithm for symmetric weight functions – Gérard Meurant

[2] V. Tomeo, E. Torrano, "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials", Linear Algebra and its Applications (2011) Volume 435, Issue 9, Pages 2314-2320, DOI=https://doi.org/10.1016/j.laa.2011.04.027 URL=http://oa.upm.es/id/eprint/8725/contents

[3] Edward B. Saff and Nikos Stylianopoulos, "Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan region", (2012) arXiv:1205.4183 [math.CV] URL=https://arxiv.org/abs/1205.4183

[4] Carmen Escribano and Antonio Giraldo and M. Asunción Sastre and Emilio Torrano, "The Hessenberg matrix and the Riemann mapping" (2011){arXiv:1107.603 [math.SP] URL=https://arxiv.org/abs/1107.6036