Jacobi Operator

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 $\ell^2(\mathbb)$. In this case it is given by

:$Jf_0 = a_0 f_1 + b_0 f_0, \quad Jf_n = a_n f_ + b_n f_n + a_ f_, \quad n>0,$

where the coefficients are assumed to satisfy

:$a_n >0, \quad b_n \in \mathbb.$

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 $p_n(x)$ of the recurrence relation

:$J\, p_n(x) = x\, p_n(x), \qquad p_0(x)=1 \text p_ (x)=0,$

is a polynomial of degree ''n'' and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector $\delta_$.

This recurrence relation is also commonly written as
:$xp_n(x)=a_p_(x) + b_n p_n(x) + a_np_(x)$

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.

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

:$zp_n(z)=\sum_^ D_ p_k(z)$

and $p_0(z)=1$. 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

: fz) = zf(z)

The zeros of the Bergman polynomial $p_n(z)$ correspond to the eigenvalues of the principal $n\times n$ submatrix of ''D''. That is, The Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator.

References

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External links

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{{DEFAULTSORT:Jacobi Operator
Operator theory
Hilbert space
Recurrence relations