The maximum-entropy approach to the solution of under determined inverse problems is studied in detail in the context of the classical moment problem. In important special cases, such as the Hausdorffmoment problem, we establish necessary and sufficient conditions for the existence of a maximum-entropy solution and examine the convergence of the resulting sequence of approximations. A number of explicit illustrations are presented. In addition to some elementary examples, we analyze the maximum-entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems. We also briefly indicate possible applications to the Lee-Yang theory of Ising models, to the summation of divergent series, and so on. The general conclusion is that maximum entropy provides a valuable approximation scheme, a serious competitor of traditional Pade-like procedures.

As a concrete example of the idea of renormalization, quantum mechanical scattering of particles by a two-dimensional delta-function potential is considered. The renormalization of the scattering cross section is carried out exactly and analytically. The calculation, free from obscuring mathematical details required for realistic field theories, may aid in making the idea of renormalization more accessible.

In this paper the principle of maximum entropy is used to predict the sum of a divergent perturbation series from the first few expansion coefficients. The perturbation expansion for the ground-state energy E(g) of the octic oscillator defined by H=p2/2+x2/2+gx8 is a series of the form E(g)∼ 1/2 +∑(−1)n+1 Angn. This series is terribly divergent because for large n the perturbation coefficients An grow like (3n)!. This growth is so rapid that the solution to the moment problem is not unique and ordinary Padé summation of the divergent series fails. A completely different kind of procedure based on the principle of maximum entropy for reconstructing the function E(g) from its perturbation coefficients is presented. Very good numerical results are obtained.

The method of finite elements converts the operator Heisenberg equations that arise from a Hamiltonian of the form // = P 2 /2+ Viq) into a set of operator difference equations on a lattice. The equal-time commutation relations are exactly preserved and thus are consistent with the requirements of unitarity. We consider general Hamiltonians of the form H(p,q) and show that the requirement of unitarity uniquely determines the operator ordering in such Hamiltonians. (The ordering procedure involves a set of orthogonal polynomials which are not widely known.) Our result shows that it is possible to treat quantum spin systems by the method of finite elements.

This paper examines underdetermined systems of partial differential equations in which the independent variables may be classical c-numbers or even quantum operators. One can view an underdetermined system as expressing the kinematic constraints on a set of dynamical variables that generate a Lie algebra. The arbitrariness in the general solution reflects the freedom to specify the dynamics of such a system.