Spectral Methods in Infinite-Dimensional Analysis

This two-volume work is devoted to the methods of the spectral theory of operators and the important role they play in infinite-dimensional analysis and its applications. Central to this study is the theory of the expansion of general eigenfunctions for families of commuting self-adjoint or normal operators. This allows a consideration of commutative models which can be applied to the representation of various commutation relations. Also included is an explanation of the theory of hypercomplex systems with locally compact bases. Applications to harmonic analysis lead to a study of the infinite-dimensional moment problem which is connected to problems of axiomatic field theory, integral representations of positive definite functions and kernels with an infinite number of variables. Infinite-dimensional elliptic differential operators are also studied. Particular consideration is given to second quantization operators and their potential perturbations, as well as Dirichlet operators. Applications to quantum field theory and quantum statistical physics are described in detail. Different variants of the theory of infinite-dimensional distributions are examined and this includes a discussion of an abstract version of white noise analysis. This work should be of interest to research mathematicians and mathematical physicists with an interest in spectral theory and its applications.