Jacobi operator

A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by a tridiagonal matrix in the standard basis given by Kronecker deltas. 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

$J\, f(1) = a(1) f(2) + b(1) f(1), \quad J\, f(n) = a(n) f(n+1) + a(n-1) f(n-1) + b(n) f(n), \quad n>1,$

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(z,n) of the recurrence relation

$J\, p(z,n) = z\, p(z,n), \qquad p(z,1)=1 \text{ and } p (z,0)=0,$

is a polynomial of degree n − 1 and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector $\delta_1(n)= \delta_{1,n}$.

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.
• three-term recurrence relationship of orthogonal polynomials, which is the basis of a set of algorithms devised to calculate Gaussian quadrature rules.

External links

• Jacobi matrix at Encyclopedia of Mathematics