A new differential calculus on a complex banach space with application to variational problems of quantum theory

Abstract

It is proved that a semilinear function on a complex banach space is not differentiable according to the usual definition of differentiability in the calculus on banach spaces. It is shown that this result makes the calculus largely inapplicable to the solution of variational problems of quantum mechanics. A new concept of differentiability called semi-differentiability is defined. This generalizes the standard concept of differentiability in a banach space and the resulting calculus is particularly suitable for optimizing real-valued functions on a complex banach space and is directly applicable to the solution of quantum mechanical variational problems. As an example of such application a rigorous proof of a generalized version of a result due to Sharma (1969) is given. In the course of this work a new concept of prelinearity is defined and some standard results in the calculus on banach spaces are extended and generalized into more powerful ones applicable directly to prelinear functions and hence yielding the standard results for linear functions as particular cases.