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.

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 ℓ 2 ( N ) {\displaystyle \ell ^{2}(\mathbb {N} )}. In this case it is given by

J f 0 = a 0 f 1 + b 0 f 0 , J f n = a n f n + 1 + b n f n + a n − 1 f n − 1 , n > 0 , {\displaystyle 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 n>0,}

where the coefficients are assumed to satisfy

a n > 0 , b n ∈ R . {\displaystyle 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 ) {\displaystyle p_{n}(x)} of the recurrence relation

J p n ( x ) = x p n ( x ) , p 0 ( x ) = 1 and p − 1 ( x ) = 0 , {\displaystyle 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 δ 1 , n {\displaystyle \delta _{1,n}}.

This recurrence relation is also commonly written as

x p n ( x ) = a n + 1 p n + 1 ( x ) + b n p n ( x ) + a n p n − 1 ( x ) {\displaystyle xp_{n}(x)=a_{n+1}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n}p_{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:

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

z p n ( z ) = ∑ k = 0 n + 1 D k n p k ( z ) {\displaystyle zp_{n}(z)=\sum _{k=0}^{n+1}D_{kn}p_{k}(z)}

and p 0 ( z ) = 1 {\displaystyle p_{0}(z)=1}. Here, D is the Hessenberg operator that generalizes the tridiagonal Jacobi operator J for this situation. Note that D is the right-shift operator on the Bergman space: that is, it is given by

[ D f ] ( z ) = z f ( z ) {\displaystyle [Df](z)=zf(z)}

The zeros of the Bergman polynomial p n ( z ) {\displaystyle p_{n}(z)} correspond to the eigenvalues of the principal n × n {\displaystyle n\times n} submatrix of D. That is, The Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator.

See also

External links