Schrr Odinger Operator in an Overfull Set

When a wave function is represented as a linear combination over an overfull set of elementary states, an ambiguity arises since such a representation is not unique. We introduce a variational principle which eliminates this ambiguity, and results in an expansion which provides the best" representation to a given Schrr odinger operator. Operational simplicity of an expansion of a wave function over a basis in the Hilbert space is an undisputable advantage for many non-relativistic quantum-mechanical computations. However, in certain cases, there are several natural" bases at one's disposal, and it is not easy to decide which is preferable. Hence, it sounds attractive to use several bases simultaneously, and to represent states as expansions over an overfull set obtained by a junction of their elements. Unfortunately, a s i s w ell known, such a representation is not unique, and lacks many convenient properties of full sets e.g., explicit formulae to compute coeecients of expansions. Because of this objection, overfull sets are used less frequently than they, perhaps, deserve. Let us consider a dense set of the wave functions 'x; k elementary states, where x is the spatial variable, and k is a label which e n umerates the states of. The parameter k can be discrete anddor continuous. We assume, for simplicity, that 'x; k are bound states of some known potentials. The density of the set means that any normalized wave function x can be approximated with a superposition of wave functions 'x; k. Consider a formal expansion of a wave function over the set : x = X k ak'x; k; 1 where P k is a summation over all the labels k integration over continuous and summation over discrete values. Expression 1 is formal, and its coeecients, ak, are not deened. The rules of computation of ak are the central point in question. In this letter, we develop a formal method which eliminates the ambiguity of the expansion 1. The main features of our approach are