A Jacobi operator, also known as Jacobi matrix, is a symmetric
linear operator acting on
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s which is given by an infinite
tridiagonal matrix
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main di ...
. 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
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
.
The name derives from a theorem from Jacobi, dating to 1848, stating that every
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
. In this case it is given by
:
where the coefficients are assumed to satisfy
:
The operator will be bounded if and only if the coefficients are bounded.
There are close connections with the theory of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
. In fact, the solution
of the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
is a polynomial of degree ''n'' and these polynomials are
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
with respect to the
spectral measure corresponding to the first basis vector
.
This recurrence relation is also commonly written as
:
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 In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...
of the
Toda lattice The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system.
It is given by a chain of particles with ...
.
* The three-term recurrence relationship of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
, 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
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
holomorphic functions
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
over some domain, then, under general circumstances, one can give that space a basis of orthogonal polynomials, the
Bergman polynomial
Bergman is a surname of German, Swedish, Dutch and Yiddish origin meaning 'mountain man', or sometimes (only in German) 'miner'.https://www.ancestry.com/name-origin?surname=bergmann
People
* Alan Bergman (born 1925), American songwriter
* Alan Be ...
s. In this case, the analog of the tridiagonal Jacobi operator is a
Hessenberg operator Hessenberg may refer to:
People:
* Gerhard Hessenberg (1874–1925), German mathematician
* Karl Hessenberg (1904–1959), German mathematician and engineer
*Kurt Hessenberg (1908–1994), German composer and professor at the Hochschule für Musik ...
– an infinite-dimensional
Hessenberg matrix
In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above ...
. The system of orthogonal polynomials is given by
:
and
. Here, ''D'' is the Hessenberg operator that generalizes the tridiagonal Jacobi operator ''J'' for this situation.
Note that ''D'' is the right-
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift ...
on the Bergman space: that is, it is given by
:
The zeros of the Bergman polynomial
correspond to the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of the principal
submatrix of ''D''. That is, The Bergman polynomials are the
characteristic polynomials for the principal submatrices of the shift operator.
References
*
External links
*
{{DEFAULTSORT:Jacobi Operator
Operator theory
Hilbert space
Recurrence relations