Jacobi operator

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 n0,

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_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.
• The theory of birth-death processes where the infinite transition matrix can be transformed into a self-adjoint Jacobi operator acting on the Hilbert space \ell^2(\mathbb{C}).

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.{{cite journal

:Df = 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.

References

• {{citation|title=Jacobi Operators and Completely Integrable Nonlinear Lattices

References

1. (2014). "Fast variants of the Golub and Welsch algorithm for symmetric weight functions in Matlab". Numerical Algorithms.
2. (1994). "Analytic birth—death processes: A Hilbert-space approach". Stochastic Processes and Their Applications.
3. (2011). "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials". Linear Algebra and Its Applications.