The Friedrichs Extension of Singular Differential Operators

410 MARLETTA AND ZETTL
We now turn our attention to Hamiltonian systems. Every 2n-th order
differential equation involving a formally symmetric operator such as M is
equivalent to a Hamiltonian system. (One possible transformation from the
2n-th order equation to a Hamiltonian system is given in (14, 15) below.)
We consider Hamiltonian systems of the form
Jy$=S(x, *) y, (11)
where J is a 2n_2n symplectic matrix and S a2n_2n symmetric matrix
given by
J= \ 0
I
&I
0 + , S(x, *)= \ S11(x, *)
S T
12
S 12
, (12)
S22(x)+ where S 11 is real and symmetric for * # R and S 22 is positive semidefinite.
We partition the solution vector y as
u
y= \ v+ ;
where u and v are vectors of length n; in case of ambiguity we shall denote
these vectors by u y and v y to indicate that they belong to y. We restrict our
attention to systems which are normal.
Definition 3. A Hamiltonian system is said to be normal on an interval
(a, b) if it has no solutions with u(x)=0 for all x #(a, b) and v(x){0
for some x #(a, b).
In particular, systems arising from differential equations w &1 My=*y
have this property.
A second property which will be of paramount importance is that of
disconjugacy.
Definition 4. The Hamiltonian system (11) is said to be disconjugate
on an interval (:, ;) (a, b) if for every interval (c, d) (:, ;) whose
endpoints are regular points of the differential equation, the boundary
value problem
Jy$=S(x, *) y, u y(c)=0 # R n , u y(d)=0 # R n
has only the trivial solution.
(13)
Notice that the property of disconjugacy is *-dependent.
Given a 2n-th order differential equation w &1 My=*y, an associated
Hamiltonian system can be defined in a number of ways. The method