The Friedrichs Extension of Singular Differential Operators

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FRIEDRICHS EXTENSION Then a self-adjoint extension L of M min is given by defining the domain of L to be D(L):=[f# D(M max):[f, j](b)&[ f, j](a)=0, j=1, ..., d]. (8) Moreover, every self-adjoint extension of M min is given in this way for suitable functions 1, ..., d. See Naimark [14] for details. Endpoints may be classified as regular or singular. We call the endpoint x=b regular if 1 p nn and p i, j ((i, j){(n, n)) together with w all lie in L 1 ((c, b), R) for some c
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
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Page 3 and 4: 406 MARLETTA AND ZETTL Suppose that
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The Friedrichs Extension of Singular Differential Operators

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