Jacobi operator

A Jacobi operator , after Carl Gustav Jakob Jacobi (1804-1851), is a symmetrical linear operator that operates on sequences and that is represented in the standard basis given by Kronecker deltas by a tridiagonal matrix , the Jacobi matrix .

Self-adjoint Jacobi operators

The most important case is that of self-adjoint Jacobi operators in the Hilbert space of the square-summable sequences over the positive integers . In this case it's through ${\displaystyle \ell ^{2}(\mathbb {N} )}$ ${\displaystyle J:\ell ^{2}(\mathbb {N} )\to \ell ^{2}(\mathbb {N} ),\,f\mapsto J\,f}$

${\displaystyle (J\,f)_{n}={\begin{cases}a_{1}f_{2}+b_{1}f_{1},&n=1,\\a_{n}f_{n+1}+a_{n-1}f_{n-1}+b_{n}f_{n},&n>1,\end{cases}}}$

given, the coefficients

${\displaystyle a_{n}>0,\quad b_{n}\in \mathbb {R} }$

fulfill. The associated operator is restricted if and only if the coefficients are. In the unrestricted case, a suitable domain must be selected.

Jacobi operators are closely related to the theory of orthogonal polynomials : The solution of the difference equation ${\displaystyle P_{n}(z)}$

${\displaystyle J\,P_{n}(z)=z\,P_{n}(z),\qquad P_{1}(z)=1,}$

is a polynomial of degree and these polynomials are orthonormal with respect to the spectral measure that belongs to the first basis vector . ${\displaystyle n}$ ${\displaystyle \delta _{1,n}}$

Applications

Jacobi operators appear in many areas of mathematics and physics. The case is known as the discrete one-dimensional Schrödinger operator. They also appear in the Lax pair of the Toda lattice . ${\displaystyle a_{n}=1}$

literature

• G. Teschl , Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs 72, Amer. Math. Soc., Providence, 2000. ISBN 0-8218-1940-2 ( free online version )