Jacobi operator

"Jacobi matrix (operator)" redirects here. For a similar term, see Jacobian matrix.

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 [math]\displaystyle{ \ell^2(\mathbb{N}) }[/math]. In this case it is given by

[math]\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\gt 0, }[/math]

where the coefficients are assumed to satisfy

[math]\displaystyle{ a_n \gt 0, \quad b_n \in \mathbb{R}. }[/math]

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 [math]\displaystyle{ p_n(x) }[/math] of the recurrence relation

[math]\displaystyle{ J\, p_n(x) = x\, p_n(x), \qquad p_0(x)=1 \text{ and } p_{-1} (x)=0, }[/math]

is a polynomial of degree n and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector [math]\displaystyle{ \delta_{1,n} }[/math].

This recurrence relation is also commonly written as

[math]\displaystyle{ xp_n(x)=a_{n+1}p_{n+1}(x) + b_n p_n(x) + a_np_{n-1}(x) }[/math]

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

[math]\displaystyle{ zp_n(z)=\sum_{k=0}^{n+1} D_{kn} p_k(z) }[/math]

and [math]\displaystyle{ p_0(z)=1 }[/math]. 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

[math]\displaystyle{ [Df](z) = zf(z) }[/math]

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

See also

• Hankel matrix

References

• {{citation|title=Jacobi Operators and Completely Integrable Nonlinear Lattices

External links

• Hazewinkel, Michiel, ed. (2001), "Jacobi matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Jacobi_matrix