An adjustable parameter is introduced into the formalism of the Rayleigh–Schrödinger perturbation theory to construct a controllable asymptotic series for the energy eigenvalues of a quantum anharmonic oscillator. Numerical results are presented, to second order in the perturbation expansion, for the ground state and the first eight excited states of the oscillator.
The Comments and Addenda section is for short communications which are not of such urgency as to justify publication in Physical Review Letters and are not appropriate for regular Articles. It includes only the following types of communications: (1) comments on papers previously published in The Physical Review or Physical Review Letters; (2) addenda to papers previously published in The Physical Review or Physical Review Letters, in which the additional information can be presented without the need for writing a complete article. Manuscripts intended for this section may be accompanied by a brief abstract for information-retrieval purposes. Accepted manuscripts will follow the same publication schedule as articles in this journal, and galleys will be sent to authors.It is shown that the curl of the velocity operator in the second quantized version of quantum hydrodynamics for interacting Bose systems is equal to zero, thus resolving the apparent discrepancies between various existing formulations.There has recently been some discussion as to the value of the commutator of the components of the velocity operator in the quantum hydrodynamics for interacting Bose systems. In a recent paper of the authors, 1 presenting a second quantized version of Landau's hydrodynamics, it was shown that this commutator was zero in apparent disagreement with the results of Landau. In more recent work 2 * 3 of the same nature, Landau's result was obtained, namely = iP'^V V k (x)-V k
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