The effect of unbounded operators on variational calculations in truncated bases

Abstract

The validity of the results of constrained variational calculations with unbounded operators in finite bases has remained unquestioned despite the mathematical difficulties pointed out by Fonte and Schiffrer. It has been show by Bassichis, Strayer and Vaughn that, as a result of these difficulties, an infinite basis calculation would yield constrained energies equal to the ground state energy, for any value of the constrained expectation value. Since similar results had not been obtained in finite basis calculations, may such calculations continue to be carried out, such as in the calculation of fission barriers. Here a detailed examination of the effect of unbounded operators in finite basis calculations is performed. It is shown that the difficulties associated with unbounded operators are not restricted to infinite bases and that, in fact, the consequences are equally severe in a finite basis. The demonstration will be made both for the one dimensional, one body model considered by Bassichis, et al., and for the full, three dimensional many body calculation currently being employed in nuclear physics.