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:
The Lax pair of the Toda lattice. The three-term recurrence relationship of orthogonal polynomials, orthogonal over a positive and finite Borel measure. Algorithms devised to calculate Gaussian quadrature rules, derived from systems of orthogonal polynomials. The theory of birth-death processes where the infinite transition matrix can be transformed into a self-adjoint Jacobi operator acting on the Hilbert space ℓ 2 ( C ) {\displaystyle \ell ^{2}(\mathbb {C} )} .
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)}
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