Jacobi operator explained

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

\ell ^{2(N)} . In this case it is given byJf _{0} = a _{0} f _{1} + b _{0} f _{0,} Jf _{n} = a _{n} f _{n+1} + b _{n} f _{n} + a _{n-1} f _{n-1} , n> 0 ,

where the coefficients are assumed to satisfy

a _{n} > 0 , b _{n} \in 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 relationJ p _{n(x)} = x p _{n(x),} p _{0(x)=1} 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 _{np} _{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} ^{n +1} ^{k =0} 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 x n submatrix of D. That is, The Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator.

See also

• Hankel matrix

Notes and References

1. 10.1007/s11075-013-9804-x . Fast variants of the Golub and Welsch algorithm for symmetric weight functions in Matlab . 2014 . Meurant . Gérard . Sommariva . Alvise . 7385259 . Numerical Algorithms . 67 . 3 . 491–506 .
2. 10.1016/j.laa.2011.04.027 . Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials . 2011 . Tomeo . V. . Torrano . E. . Linear Algebra and Its Applications . 435 . 9 . 2314–2320 . free .
3. Saff . Edward B. . Stylianopoulos . Nikos . 1205.4183 . 10.1007/s11785-012-0252-8 . 1 . Complex Analysis and Operator Theory . 3147709 . 1–24 . Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions . 8 . 2014.
4. Escribano . Carmen . Giraldo . Antonio . Sastre . M. Asunción . Torrano . Emilio . 1107.6036 . 10.1007/s10444-012-9291-y . 3-4 . Advances in Computational Mathematics . 3116040 . 525–545 . The Hessenberg matrix and the Riemann mapping function . 39 . 2013.