## Boundary value problems and symplectic algebra

Norrie Everitt and Lawrence Markus published the monograph

*Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators*in 1999. This work was published by the American Mathematical Society as volume 61 in their Mathematical Surveys and Monographs series. We give below a version of the Preface to the monograph:The origins of this monograph lie, firstly, in the pioneering contributions from H Weyl , J von Neumann, M H Stone, E C Titchmarsh, K Kodaira to the theory of linear differential operators in Hilbert function spaces and, secondly, in the significant contributions made by the Ukrainian (former Soviet Union) mathematicians M G Krein, M A Naimark and I M Glazman to the study of boundary value problems for linear, ordinary quasi-differential equations on any real interval.

The results of Glazman in his seminal memoir of 1950, influenced by both Krein and Naimark, led to the now-named GKN theorem within the general theory of quasi-differential operators (which include and generalise the classical linear ordinary differential operators) in Hilbert function spaces. the significant contribution from Glazman, mirrored in part by the work (also in 1950) of Kodaira, led to the then new formulation of boundary conditions required to construct self-adjoint differential operators, representing the boundary value problem. The original GKN theorem is stated for real-valued, thereby necessarily of even order, quasi-differential expressions; the theorem gives an elegant, necessary and sufficient condition for Lagrange symmetric differential expressions to generate self-adjoint operators in the appropriate Hilbert space of functions on the prescribed real interval.

The Glazman idea is to represent the homogeneous boundary conditions in terms of the skew-symmetric, sesquilinear form associated with the quasi-differential expression and the corresponding Green's formula: the quasi-differential expressions arc now known to define a real symplectic space, and the boundary conditions to correspond to Lagrangian subspaces of this symplectic space, as recently recognised and realised by the current authors. The properties of these real symplectic spaces, and their geometry and symplectic linear algebra, have long been advanced by mathematicians and physicists in a number of different applications: in particular in Lagrangian analytical dynamics and quantum theory.

The original GKN theory was confined to the real-valued, quasi-differential expressions of arbitrary even order. However complex-valued quasi-differential expressions, of arbitrary (positive) integer order, had been studied earlier by Halperin and Shin, and later by Everitt and Zettl. In the years following the untimely death of Glazman in 1968 these complex expressions have been extensively studied, with particular reference to the Lagrange symmetric (formally self-adjoint) expressions. This formulation of the GKN Theorem has consequently been extended to these complex quasi-differential expressions of arbitrary integer order; however this extension has required the introduction and study of linear complex symplectic geometries and the algebra of their Lagrangian subspaces, as defined and described in these pages. The consequences of this study are to be seen in the contents of this monograph.

Two special comments are called for in respect of these complex symplectic spaces:

1. The complex spaces have a much richer structure and range of properties in comparison with the real spaces. Real symplectic spaces exist in even dimensions only, and moreover there is a unique real space (up to symplectic isomorphism) in each such even dimension. Every real symplectic space can be complexified to a complex symplectic space of the same even (complex) dimension. However there exist even-order complex symplectic spaces that are not the complexification of any real space, and there exist different complex symplectic spaces of each odd integer order.

2. The complex, Lagrange symmetric, quasi-differential expressions, of arbitrary positive integer order

It can be argued that complex symplectic spaces have richer structures in order to support the extensive properties of complex quasi-differential expressions; vice versa there is a case to state that these complex differential expressions force the structure of the complex symplectic spaces to exist in order to support their properties.
2. The complex, Lagrange symmetric, quasi-differential expressions, of arbitrary positive integer order

*n*, also have additional structures in comparison with the corresponding real expressions. the most significant property, in this respect, is that the complex expressions lead to minimal, closed symmetric operators, defined in the appropriate Hilbert function space, which can have unequal deficiency indices: these indices are now re-interpreted as algebraic invariants of the corresponding complex symplectic space. Of course, such a minimal symmetric operator has self-adjoint extensions if and only if the two deficiency indices are of the same value, say a non-negative integer*d*, which is less than or equal to the order*n*. In fact, an informal paraphrase of our new version of the GKN Theorem asserts:However in our detailed analysis of the kinds of boundary conditions that can occur we partition the basic interval into left and right sub-intervals, on each of which the restricted differential operator may have unequal deficiency indices. Thus the full range for the deficiency indices, equal or not, plays an important role in the theory.Each such self-adjoint operator is specified explicitly by a Lagrangian d-space within the corresponding boundary complex symplectic2d-space, and conversely each Lagrangian d-space corresponds to exactly one such self-adjoint operator.

The two main and significant consequences of writing this research monograph are:

1. There is now a complete and connected account of the geometric and algebraic structure of real and complex symplectic spaces and their Lagrangian subspaces, for all integer orders, with special attention to the algebraic properties of direct sum decompositions such as are relevant for the study of boundary conditions, especially with regard to properties of separation or coupling at the boundary endpoints.

2. There is a complete account of the canonical form of all possible symmetric boundary conditions (with respect to separation or coupling at the endpoints) for the extended GKN theory of Lagrange symmetric, linear, quasi-differential expressions (real or complex) of all integer orders on arbitrary real intervals.

In addition to this main text there are two substantial appendices. the first deals with the canonical form of classical ordinary differential expressions when these are considered as quasi-differential expressions and then settles certain technical questions concerning adjoint operators; the second treats the problems of the complexification of real symplectic spaces, and the analysis of self-adjoint operators which are non-real yet arise from real differential expressions.
2. There is a complete account of the canonical form of all possible symmetric boundary conditions (with respect to separation or coupling at the endpoints) for the extended GKN theory of Lagrange symmetric, linear, quasi-differential expressions (real or complex) of all integer orders on arbitrary real intervals.

In all these areas the authors have made significant and extensive new contributions, in addition to re-organising established theories into a satisfying synthesis with the results within this monograph. As an illustration of our approach and of some of the new results, we offer here two very specific and explicit findings:

(i) the balanced intersection principle provides an algebraic criterion for describing and classifying the divers kinds of self-adjoint boundary conditions for quasi-differential expressions of arbitrary integer order, and for all boundary value problems whether regular on compact intervals or singular on general intervals. In particular the coupling grade is defined for each Lagrangian

For singular problems this is not necessarily the case; however there is an arithmetic formula relating the number of separated boundary conditions at each of the two endpoints with the invariants of the left and right endpoint complex symplectic spaces. For example, consider a Lagrange symmetric real quasi-differential expression of order four on the closed half-line [0, ∞). We find that the common deficiency index

(ii) Lagrange symmetric quasi-differential expressions that are real can determine self-adjoint operators that are real (definable by real boundary conditions) or else complex operators that are non-real. An investigation of this phenomenon is conducted in Appendix B, where the complexification of real symplectic spaces, and the associated concept of self-conjugate Lagrangian subspace, are described in great detail.

We provide an affirmative answer to a long-standing open question concerning the existence of real differential expressions of even order ≥ 4, for which t here are non-real self-adjoint differential operators specified by strictly separated boundary conditions, i.e. complex Lagrangian subspaces which are not self-conjugate and which have coupling grade zero; in fact, we prove the existence of such Lagrangian subspaces of every possible prescribed coupling grade. This is somewhat surprising because it is well known that for order *d*-space (and hence for the corresponding self-adjoint operator) and from this we deduce the minimal number of coupled boundary conditions necessary in the specification of the operator domain. For a regular problem of arbitrary positive order, on a compact interval, there is always the same number of separated boundary conditions at the left endpoint as at the right endpoint of the interval (assuming that minimal coupling is employed).For singular problems this is not necessarily the case; however there is an arithmetic formula relating the number of separated boundary conditions at each of the two endpoints with the invariants of the left and right endpoint complex symplectic spaces. For example, consider a Lagrange symmetric real quasi-differential expression of order four on the closed half-line [0, ∞). We find that the common deficiency index

*d*can take the values 2, 3 or 4, which generalises the limit-point and limit-circle classifications that Weyl defined for second-order differential expressions. As an indication of the explicit nature of our calculations and tabulations, we mention that it is then possible to have three (independent) boundary conditions, when*d*= 3, to define a self-adjoint operator, with one separated at the left end and two coupling the ends, but it is impossible to define a self-adjoint operator by one separated condition at the left end and two separated at the right end.(ii) Lagrange symmetric quasi-differential expressions that are real can determine self-adjoint operators that are real (definable by real boundary conditions) or else complex operators that are non-real. An investigation of this phenomenon is conducted in Appendix B, where the complexification of real symplectic spaces, and the associated concept of self-conjugate Lagrangian subspace, are described in great detail.

*n*= 2 strictly separated boundary conditions can produce only real operators (that is, any such given complex boundary conditions can always be replaced by real boundary conditions). Our analyses and examples are entirely explicit for regular problems on compact intervals; moreover corresponding explicit results also hold in the general singular case.

In undertaking this project we have reviewed the theory of both differential and quasi-differential expressions and have assembled all relevant information in the opening sections of this monograph to provide a convenient source of reference. In particular we have put together details of the connections between classical differential expressions and the extended class of quasi-differential expressions .

In respect of symmetric boundary problems for these differential expressions, and the associated self-adjoint differential operators, there is complete generality; regular problems on compact intervals and the more general singular problems are all treated in full detail. From the algebraic data all the classification results for boundary conditions then follow; thereby the infinite dimensional functional analysis is reduced to finite dimensional linear symplectic algebra.

The introduction of these algebraic and geometric methods has led to the discovery of new kinds of qualitative insight into the topology of the boundary value problem in terms of the Lagrange-Grassmannian manifold.

The axiomatic formulation of these mathematical structures leads immediately to applications for other types of boundary value problems such as the multi-interval or interfacial conditions of the multi-particle systems of quantum mechanics, or the general theory of linear elliptic partial differential equations; these applications depend on the extension of the ideas considered in t his monograph to infinite dimensional, complex symplectic spaces.

In concluding this work we have to survive a disappointment. It had been our hope at the start of these labours that the algebra of complex symplectic spaces would throw new light on the "deficiency index conjecture" for complex quasi-differential operators. This has not been the case and the so-called range conjecture, formulated precisely in this work, remains unsolved. We have little doubt that the conjecture is true. While our analysis and classification of symmetric boundary conditions do not rest on the validity of this conjecture, we have occasionally used it (with appropriate warnings) to give insight and guidance into the search for new properties and interrelations among classes of such boundary value problems.