A characterization of positive self-adjoint extensions and its application to ordinary differential operators

A new characterization of the positive self-adjoint extensions of symmetric operators, , is presented, which is based on the Friedrichs extension of a direct sum decomposition of domain of the adjoint and the boundary mapping of . In applying this result to ordinary differential equations, we characterize all positive self-adjoint extensions of symmetric regular differential operators of order in terms of boundary conditions.

     

Characterization of domains of self-adjoint ordinary differential operators

The GKN (Glazman, Krein, Naimark) Theorem characterizes all self-adjoint realizations of linear symmetric (formally self-adjoint) ordinary differential equations in terms of maximal domain functions. These functions depend on the coefficients and this dependence is implicit and complicated. In the regular case an explicit characterization in terms of two-point boundary conditions can be given. In the singular case when the deficiency index d is maximal the GKN characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter λ. In the much more difficult intermediate cases, not all solutions contribute to the singular self-adjoint conditions. In 1986 Sun found a representation of the self-adjoint singular conditions in terms of certain solutions for nonreal values of λ. In this paper we give a representation in terms of certain solutions for real λ. This leads to a classification of solutions as limit-point (LP) or limit-circle (LC) in analogy with the celebrated Weyl classification in the second-order case. The LC solutions contribute to the singular boundary conditions, the LP solutions do not. The advantage of using real λ is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.     

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.     

Weak and strong extensions of ordinary differential operators

Let A be a second order differential operator with positive leading term defined on an interval J of R. In this paper we study conditions for the equality D₀(A) = D₁(A) to hold. Here D₀(A) and D₁(A) are the domains of the minimal and maximal extensions of A respectively. Under the general assumption that $\text{A(1)}\text{and}\text{A}{}^1\text{(1)}$ are bounded above it is proven that under certain conditions D₀(A) = D₁(A) if functions which are constant near the boundaries of J are in $\text{D}{}_0\text{(A) ∩ D}{}_0\text{(A}{}^1\text{)}$ whenever they are in $\text{D}{}_1\text{(A) ∩ D}{}_1\text{(A}{}^1\text{)}$. In particular if A is formally selfadjoint and 1 ?D₁(A) then D₁(A) = D₀(A) if and only if 1 ?D₀(A). When the measure of J is infinite at both ends D₀(A) is always equal to D₁(A). This fact is used to show that the leading term of A as well as its terminal coefficient can be chosen arbitrarily (although not independently of one another) in such a way that the equality D₀ = D₁ holds.