Elliptic partial differential operators and symplectic algebra

W Everitt, L Markus

Research output: Contribution to journalArticle

12 Citations (Scopus)


This investigation introduces a new description and classification for the set of all self-adjoint operators (not just those defined by differential boundary conditions) which are generated by a linear elliptic partial differential expression A (x, D) = Sigma(0less than or equal to\s\less than or equal to2m) a(s)(x)D-s for all x is an element of Omega in a region Omega, with compact closure (&UOmega;) over bar and C-infinity-smooth boundary partial derivativeOmega, in Euclidean space E-r (r greater than or equal to 2). The order 2m greater than or equal to 2 and the spatial dimension r greater than or equal to 2 are arbitrary. We assume that the coefficients a(s) is an element of C-infinity ((&UOmega;) over bar) are complex-valued, except real for the highest order terms (where \s\ = 2m) which satisfy the uniform ellipticity condition in (&UOmega;) over bar. In addition, A((.), D) is Lagrange symmetric so that the corresponding linear operator A, on its classical domain D(A) := C-0(infinity)(Omega) subset of L-2(Omega), is symmetric; for example the familiar Laplacian Delta and the higher order polyharmonic operators Delta(m). Through the methods of complex symplectic algebra, which the authors have previously developed for ordinary differential operators, the Stone-von Neumann theory of symmetric linear operators in Hilbert space is reformulated and adapted to the determination of all self-adjoint extensions of A on D(A), by means of an abstract generalization of the Glazman-Krein-Naimark (GKN) Theorem. In particular the authors construct a natural bijective correspondence between the set {T} of all such self-adjoint operators on domains D(T) superset of D(A), and the set {L} of all complete Lagrangian subspaces of the boundary complex symplectic space S = D (T-1) / D (T-0), where T-0 on D (T-0) and T-1 on D (T-1) are the minimal and maximal operators, respectively, determined by A on D(A) C L2(Q). In the case of the elliptic partial differential operator A, we verify D(T-0) = (W) over circle (2m)(Omega) and provide a novel definition and structural analysis for D(T-1) = W-A2m(Omega), which extends the GKN-theory from ordinary differential operators to a certain class of elliptic partial differential operators. Thus the boundary complex symplectic space S = W-A2m(Omega) / (W) over circle (2m)(Omega) effects a classification of all self-adjoint extensions of A on D(A), including those operators that are not specified by differential boundary conditions, but instead by global (i.e. non-local) generalized boundary conditions. The scope of the theory is illustrated by several familiar, and other quite unusual, self-adjoint operators described in special examples.
Original languageEnglish
Pages (from-to)1-+
JournalMemoirs of the American Mathematical Society
Issue number770
Publication statusPublished - 1 Jan 2003


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