Jacobi operator

Last updated January 30, 2024

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.

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 bounded.

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 principal $n\times n$ submatrix of D. That is, The Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator.

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References

↑ Meurant, Gérard; Sommariva, Alvise (2014). "Fast variants of the Golub and Welsch algorithm for symmetric weight functions in Matlab" (PDF). Numerical Algorithms. 67 (3): 491–506. doi:10.1007/s11075-013-9804-x. S2CID 7385259.
↑ Tomeo, V.; Torrano, E. (2011). "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials". Linear Algebra and Its Applications. 435 (9): 2314–2320. doi: 10.1016/j.laa.2011.04.027 .
↑ Saff, Edward B.; Stylianopoulos, Nikos (2012). "Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions". arXiv: 1205.4183 .{{cite journal}}: Cite journal requires |journal= (help)
↑ Escribano, Carmen; Giraldo, Antonio; Asunción Sastre, M.; Torrano, Emilio (2011). "The Hessenberg matrix and the Riemann mapping". arXiv: 1107.6036 .{{cite journal}}: Cite journal requires |journal= (help)

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

External links

"Jacobi matrix", Encyclopedia of Mathematics , EMS Press, 2001 [1994]

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[1] Meurant, Gérard; Sommariva, Alvise (2014). "Fast variants of the Golub and Welsch algorithm for symmetric weight functions in Matlab" (PDF). Numerical Algorithms. 67 (3): 491–506. doi:10.1007/s11075-013-9804-x. S2CID 7385259.

[2] Tomeo, V.; Torrano, E. (2011). "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials". Linear Algebra and Its Applications. 435 (9): 2314–2320. doi: 10.1016/j.laa.2011.04.027 .

[3] Saff, Edward B.; Stylianopoulos, Nikos (2012). "Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions". arXiv: 1205.4183 .{{cite journal}}: Cite journal requires |journal= (help)

[4] Escribano, Carmen; Giraldo, Antonio; Asunción Sastre, M.; Torrano, Emilio (2011). "The Hessenberg matrix and the Riemann mapping". arXiv: 1107.6036 .{{cite journal}}: Cite journal requires |journal= (help)

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