# Duality relations in the auxiliary field method

Abstract : The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$ identical particles with a mass $m$ are functions of the various radial quantum numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$ of these eigenenergies, depending on a principal quantum number $Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the potentials (independent of $m$ and $N$) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$.
Document type :
Journal articles

http://hal.in2p3.fr/in2p3-00564101
Contributor : Emmanuelle Vernay <>
Submitted on : Tuesday, February 8, 2011 - 8:28:50 AM
Last modification on : Thursday, November 19, 2020 - 12:58:54 PM

### Citation

B. Silvestre-Brac, C. Semay. Duality relations in the auxiliary field method. Journal of Mathematical Physics, American Institute of Physics (AIP), 2011, 52, pp.052107. ⟨10.1063/1.3589959⟩. ⟨in2p3-00564101⟩

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