Series Expansions

Michael A. Taylor

Jürgen König, Alfred Hucht

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

Jürgen König

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

R. Shankar

R. A. Corns

Two series representations of the propagator associated with the quantum anharmonic-oscillator are developed. A closed form for the Dyson series expansion of the propagator is obtained by using a phase-space perturbation technique. For Abelian interactions the Dyson series can be rearranged into an exponentiated connected-graph series. This representation is structurally similar to the cluster expansions of propagators associated with perturbations of the kinetic-energy Hamiltonian. The connected-graph series is particularly useful for the development of semiclassical expansions such as the WKB expansion. The analytic structure in the physical parameters such as mass, time and Planck's constant is readily extracted from the connected-graph series because of the explicit closed-form formula for the series.

Stephen J. Gustafson, Israel Michael Sigal