Jacobi-Operator

A Jacobi operator, according to Carl Gustav Jacob Jacobi, is a symmetric linear operator operating on consequences and is represented in the given by Kronecker deltas standard basis by a tridiagonal matrix.

Self-adjoint Jacobi operators

The most important case is that of self-adjoint Jacobi operators in the Hilbert space of quadratsummierbaren consequences on the positive integers. In this case, by

Where, the coefficients

. meet The corresponding operator is bounded if and only if the coefficients. In the unrestricted case, a suitable definition range must be selected.

Jacobi operators are closely linked to the theory of orthogonal polynomials: The solution of the difference equation

Is a polynomial of degree and these polynomials are orthonormal with respect to the Spektralmaßes belonging to the first base vector.

Applications

Jacobi operators appear in many fields of mathematics and physics. The case is known as a discrete one-dimensional Schrödinger operator. They occur also in the Lax pair of Toda lattice.