One of the masters in the differential equations community, the late F.V. Atkinson contributed seminal research to multiparameter spectral theory and Sturm-Liouville theory. His ideas and techniques have long inspired researchers and continue to stimulate discussion. With the help of co-author Angelo B. Mingarelli, Multiparameter Eigenvalue Problems: Sturm-Liouville Theory reflects much of Dr. Atkinsona (TM)s final work.
After covering standard multiparameter problems, the book investigates the conditions for eigenvalues to be real and form a discrete set. It gives results on the determinants of functions, presents oscillation methods for Sturm-Liouville systems and other multiparameter systems, and offers an alternative approach to multiparameter Sturm-Liouville problems in the case of two equations and two parameters. In addition to discussing the distribution of eigenvalues and infinite limit-points of the set of eigenvalues, the text focuses on proofs of the completeness of the eigenfunctions of a multiparameter Sturm-Liouville problem involving finite intervals. It also explores the limit-point, limit-circle classification as well as eigenfunction expansions.
A lasting tribute to Dr. Atkinsona (TM)s contributions that spanned more than 40 years, this book covers the full multiparameter theory as applied to second-order linear equations. It considers the spectral theory of multiparameter problems in detail for both regular and singular cases.
F.V. Atkinson was a professor emeritus of mathematics at the University of Toronto. A Fellow of the Royal Society of Canada and an Honorary Fellow of the Royal Society of Edinburgh, Dr. Atkinson was awarded the Makdougall-Brisbane Prize of the Royal Society of Edinburgh for his enduring paper on limit-n criteria of integral type. He published more than 100 papers on subjects ranging from the theory of the Riemann zeta function to operator theory. He earned his Ph.D. from the University of Oxford, under the guidance of E.C. Titchmarsh. Angelo B. Mingarelli is a professor of mathematics at Carleton University. He previously taught at the Pennsylvania State University and the University of Ottawa. Dr. Mingarelli has been an NSERC University Research Fellow for many years and has won numerous awards for excellence in teaching. He earned his Ph.D. from the University of Toronto, under the supervision of F.V. Atkinson.
Preliminaries and Early History Main results of Sturm-Liouville theory General hypotheses for Sturm-Liouville theory Transformations of linear second-order equations Regularization in an algebraic case The generalized Lame equation Klein's problem of the ellipsoidal shell The theorem of Heine and Stieltjes The later work of Klein and others The Carmichael program Some Typical Multiparameter Problems The Sturm-Liouville case The diagonal and triangular cases Transformations of the parameters Finite difference equations Mixed column arrays The differential operator case Separability Problems with boundary conditions Associated partial differential equations Generalizations and variations The half-linear case A mixed problem Definiteness Conditions and the Spectrum Introduction Eigenfunctions and multiplicity Formal self-adjointness Definiteness Orthogonalities between eigenfunctions Discreteness properties of the spectrum A first definiteness condition, or "right-definiteness" A second definiteness condition, or "left-definiteness" Determinants of Functions Introduction Multilinear property Sign-properties of linear combinations The interpolatory conditions Geometrical interpretation An alternative restriction A separation property Relation between the two main conditions A third condition Conditions (A), (C) in the case k = 5 Standard forms Borderline cases Metric variants on condition (A) Oscillation Theorems Introduction Oscillation numbers and eigenvalues The generalized Prufer transformation A Jacobian property The Klein oscillation theorem Oscillations under condition (B), without condition (A) The Richardson oscillation theorem Unstandardized formulations A partial oscillation theorem Eigencurves Introduction Eigencurves Slopes of eigencurves The Klein oscillation theorem for k = 2 Asymptotic directions of eigencurves The Richardson oscillation theorem for k = 2 Existence of asymptotes Oscillation Properties for Other Multiparameter Systems Introduction An example Local definiteness Sufficient conditions for local definiteness Orthogonality Oscillation properties The curve I = f(I",m) The curve I" = g(I , n) Distribution of Eigenvalues Introduction A lower order-bound for eigenvalues An upper order-bound under condition (A) An upper bound under condition (B) Exponent of convergence Approximate relations for eigenvalues Solubility of certain equations The Essential Spectrum Introduction The essential spectrum Some subsidiary point-sets The essential spectrum under condition (A) The essential spectrum under condition (B) Dependence on the underlying intervals Nature of the essential spectrum The Completeness of Eigenfunctions Introduction Green's function Transition to a set of integral equations Orthogonality relations Discussion of the integral equations Completeness of eigenfunctions Completeness via partial differential equations Preliminaries on the case k = 2 Decomposition of an eigensubspace Completeness via discrete approximations The one-parameter case The finite-difference approximation The multiparameter case Finite difference approximations Limit-Circle, Limit-Point Theory Introduction Fundamentals of the Weyl theory Dependence on a single parameter Boundary conditions at infinity Linear combinations of functions A single equation with several parameters Several equations with several parameters More on positive linear combinations Further integrable-square properties Spectral Functions Introduction Spectral functions Rate of growth of the spectral function Limiting spectral functions The full limit-circle case Appendix on Sturmian Lemmas Bibliography Index Research problems and open questions appear at the end of each chapter.
The book reads well and is accessible to everyone with a background in one-parameter Sturm-Liouville theory. The second author is successful in maintaining Atkinson's admirable style of writing. --Hans W. Volkmer, Mathematical Reviews, Issue 2011k
F.V. Atkinson, Angelo B. Mingarelli
MULTIPARAMETER EIGENVALUE PROB
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