Sobre o método variacional em problemas espectrais

Abstract

The study of operator theory is important in the vast branches of exact sciences, where many processes can be described by an operator, whose properties vary according to the space in which it is defined, among which is Hilbert space1. The theory of linear operators on Hilbert space is mainly oriented to applications of the spectral properties of integral and differential operators, i.e., in the analysis of the equation Lu = Àu, (1) where L is a differential operator. In the work [3] "Theory of Linear Operators in Hilbert Space", by N. I. Akhiezer and I. M. Glazman, the properties of unbounded linear operators that are defined on a restricted domain of Hilbert space H and whose domains of these operators are dense on H are studied. In the work [5] "Introduction to Spectral Theory in Hilbert Space", by G. Helmberg, one studies the properties of linear operators (not necessarily bounded) in the concrete Hilbert space L2, and whose domains of these operators are a linear variety. The work [9] "Applications of functional analysis and operator theory", by V. Hutson, J. Pym, M. Cloud, studies the applications and properties of compact and self-adjoint ordinary differential operators on Banach spaces2 and Hilbert subspaces. In general in the above highlighted works the applications and properties of operators defined on a subspace of Hilbert space H are studied,