Functional Analysis: Hilbert Space

Lawrence Paul Horwitz, Yosef Strauss

Liwei Liu

For the purpose of making abstract Hilbert spaces easily comprehensible,represents comparison between N-dimension vector space and quantum mechanics Hilbert space,from space composing and space variable calculation to space coordinate transformation and so on. A conclusion that two spaces are comparable can be drawn.Various calculation of Hilbert space composed of wave function is easily comprehensible and grasped.

Pankaj Sharan

Peter Krämer, D. Schenzle

The relation between the Hilbert space H of quantum mechanics and the Hilbert space F of entire analytic functions introduced by Bargmann is used to derive unitary projective representations of the group SL(2,R) of linear canonical transformations. These representations are applied to an analysis of various nuclear models. (auth)

Terence William Barrett

A method of defining structural information in Hilbert space is described. Instead of commencing an information theory analysis with a simple signal and progressing to an n-dimensional information space, the reverse procedure can be followed: the signal is defined completely in both real and imaginary spaces, and is then represented as a dimensionless number in Hilbert space. Thus, a signal can be analyzed into a Hilbert space measure of information. This measure has similarities to the relativity of states of Everett's (1957) reformulation of quantum mechanics involving observer-observed interaction. Whereas Everett's theory deals with the totality of all possible ways in which a state function can be decomposed into the sum of products of state functions for subsystems of the overall system, the theory presented here deals with the totality of possible ways in which a Hilbert space measure of signal state can be decomposed by signal definition into reciprocally related measures of the state for subsystems of the overall system. In the overall system presented here, a signal is given a concise representation either as a quadratic form on a Hilbert space, or as a Fourier coefficients in isomorphic function space. The total information content of a signal qualified as a dimensionless number in Hilbert space may be given a representation in familiar units by the use of such methods. A primary postulate of quantum mechanics is obeyed in that all physically relevant information about a system is derivable from the knowledge of the state function of the system. Thus, a signal is defined firstly as an information measure, and secondly in the familiar units of frequency, bandwidth, midperiod, and duration.