Perturbation Of Isolated Eigenvalues Of Singular Differential Operators

Perturbation of Isolated Eigenvalues of Singular Differential Operators

The objective of book is to investigate the results of limit circle case/limit point case of singular Sturm-Liouville differential operators about their spectrum and invariance under perturbation that arise in quantum mechanics. The studies of ordinary differential operators of any order and dimension have been motivated by the Herman Weyl's selected work on general singular ordinary differential expressions together with the development in quantum mechanics. The Sturm-Liouville differential equation is one of the particular forms of the general singular ordinary differential expression and it forms generalization of well known differential equations such as Bessel, Laguerre, Hermite and Legendre's differential equations which are found to have applications in several branches of mathematical physics.

Perturbation theory for linear operators

Ordinary Differential Operators

In 1910 Herman Weyl published one of the most widely quoted papers of the 20th century in Analysis, which initiated the study of singular Sturm-Liouville problems. The work on the foundations of Quantum Mechanics in the 1920s and 1930s, including the proof of the spectral theorem for unbounded self-adjoint operators in Hilbert space by von Neumann and Stone, provided some of the motivation for the study of differential operators in Hilbert space with particular emphasis on self-adjoint operators and their spectrum. Since then the topic developed in several directions and many results and applications have been obtained. In this monograph the authors summarize some of these directions discussing self-adjoint, symmetric, and dissipative operators in Hilbert and Symplectic Geometry spaces. Part I of the book covers the theory of differential and quasi-differential expressions and equations, existence and uniqueness of solutions, continuous and differentiable dependence on initial data, adjoint expressions, the Lagrange Identity, minimal and maximal operators, etc. In Part II characterizations of the symmetric, self-adjoint, and dissipative boundary conditions are established. In particular, the authors prove the long standing Deficiency Index Conjecture. In Part III the symmetric and self-adjoint characterizations are extended to two-interval problems. These problems have solutions which have jump discontinuities in the interior of the underlying interval. These jumps may be infinite at singular interior points. Part IV is devoted to the construction of the regular Green's function. The construction presented differs from the usual one as found, for example, in the classical book by Coddington and Levinson.