Upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection

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Abstract
Renormalization is a method often used to approximate the eigenvalues of a Hamiltonian that cannot be solved exactly. It consists of splitting the Hamiltonian into a solvable part and a remainder which is then minimized. The inner projection technique, first introduced by Löwdin in the sixties, was developed to bracket the eigenvalues between lower and upper bounds. Combining renormalization and Löwdin’s inner projection yielded the so-called ‘‘renormalized inner projection technique.’’ In this study, this method will be applied to the quartic, sextic, and octic anharmonic oscillators. Lower and upper energy bounds are obtained for finite values of the coupling constant as well as for the infinite case. The relation between the renormalized inner projection and perturbation theory will also be discussed. Another feature of this study is the importance of symbolic computation in allowing us to manipulate expressions with unevaluated parameters and to perform calculations in rational arithmetics or high decimal precision. Thus Löwdin’s rational approximants can be expressed explicitly as rational fractions in terms of the coupling constant and values for the limit constant can be obtained with amazing high accuracy, namely, 62, 33, and 21 decimal places for the quartic, sextic, and octic oscillator, respectively.

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