Invariant Formulation
Definition The operators, are called equivalent if there is a gauge transformation that takes one to the other:
BK-factorization is then pure algebraic procedure which allows to construct explicitly a factorization of an arbitrary order LPDO in the form
with first-order operator where is an arbitrary simple root of the characteristic polynomial
Factorization is possible then for each simple root iff
for
for
for
and so on. All functions are known functions, for instance,
and so on.
Theorem All functions
are invariants under gauge transformations.
Definition Invariants are called generalized invariants of a bivariate operator of arbitrary order.
In particular case of the bivariate hyperbolic operator its generalized invariants coincide with Laplace invariants (see Laplace invariant).
Corollary If an operator is factorizable, then all operators equivalent to it, are also factorizable.
Equivalent operators are easy to compute:
and so on. Some example are given below:
Read more about this topic: Invariant Factorization Of LPDOs
Famous quotes containing the word formulation:
“You do not mean by mystery what a Catholic does. You mean an interesting uncertainty: the uncertainty ceasing interest ceases also.... But a Catholic by mystery means an incomprehensible certainty: without certainty, without formulation there is no interest;... the clearer the formulation the greater the interest.”
—Gerard Manley Hopkins (18441889)