Bounded matrix elements for the quartic-anharmonic oscillator

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Abstract
A method based on quantum-mechanical sum rules is presented in which matrix elements of the displacement and the square of the displacement for a general one-dimensional oscillator are expressed directly in terms of the eigenenergies and the potential parameters. Matrix elements of higher powers of the displacement can then be calculated through an exact hypervirial relation. Two important features of the present method, the rapid convergence and the result that successive approximations are alternately upper and lower bounds to the matrix elements, make the method useful in those cases for which accurate eigenenergies are known. Sample calculations for the quartic-anharmonic oscillator are presented and compared with matrix elements obtained by other iterative schemes.