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## Second Order Operator and the Bilinear Concomitant

Let us extend our considerations from linear differential operators of first order to those of second order. To do this, let us find the adjoint of a second order operator. The given operator consists of

(i)
the differential operator

(ii)
the domain on which it operates,

where and are two homogeneous boundary conditions,

 (43)

The and are given constants not to be confused with the functions and . The task is to find the adjoint of the given operator, namely FIND
(i)
(ii)
such that

for all and all . The left-hand side of this equation is given, and it is

In order to have the derivatives act on the function , one does an integration by parts twice on the first term, and once on the second term. The result is

The bilinear expression is called the bilinear concomitant or the conjunct of and . Thus we have

 (44)

This important integral identity is the one-dimensional version of Green's identity. Indeed, it relates the behavior of and in the interior of to their values on the boundary, here and . It is an extension of the integrated Lagrange identity, Eq.(3.16), from formally self-adjoint second order operators to generic second order operators. Observe that when

becomes formally self-adjoint whenever the coefficient functions , and are real. In this circumstance is the Sturm-Liouville operator and the bilinear concomitant reduces to

which is proportional to the Wronskian determinant of and . The construction of from is based on the requirement that

This means that the bilinear concomitant evaluated at the endpoints must vanish,

This is a compatibility condition between the given boundary conditions, Eq.(4.3),

This means that any two of the three sets of conditions implies the third:
1. (II) and (III) imply (I).
2. (III) and (I) imply (II).
3. (I) and (II) imply (III).
The problem of obtaining the adjoint boundary conditions in explicit form,

 (45)

is a problem in linear algebra. One must combine the given boundary conditions, Eq.(4.3) with the compatibility condition (I) to obtain the coefficients in Eq.(4.5).

Next: Green's Function and Its Up: The Adjoint of an Previous: Adjoint Boundary Conditions   Contents   Index
Ulrich Gerlach 2010-12-09