State (functional analysis)

Abstract: We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Neel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

Abstract: The initial state of an unforced linear system is output admissible with respect to a constraint set Y if the resulting output function satisfies the pointwise-in-time condition y(t) in Y, t>or=0. The set of all possible such initial conditions is the maximal output admissible set O/sub infinity /. The properties of O/sub infinity / and its characterization are investigated. In the discrete-time case, it is generally possible to represent O/sub infinity / or a close approximation of it, by a finite number of functional inequalities. Practical algorithms for generating the functions are described. In the continuous-time case simple representations of the maximal output admissible set are not available, however, it is shown that the discrete-time results may be used to obtain approximate representations. >

Abstract: It is attempted to formulate the properties of localized states on the basis of natural invariance requirements. Chief of these is that a state, localized at a certain point, becomes, after a translation, orthogonal to all the undisplaced states localized at that point. It is found that the required properties uniquely define the set of localized states for elementary systems of non-zero mass and arbitrary spin. The localized functions belong to a continuous spectrum of an operator which it is natural to call the position operator. This operator has automatically the property of preserving the positive energy character of the wave function to which it is applied (and it should be applied only to such wave functions). It is believed that the development here presented may have applications in the theory of elementary particles and of the collision matrix.