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.
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
This recurrence relation is also commonly written as 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: 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.