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 $E^{(p)}(m^{\prime };Q^{\prime })$ 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 ${\epsilon }^{(N)}(m;\{n_i,l_i\})$.

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Cited by in corpus (10)

• Numerical tests of the envelope theory for few-boson systems
• Spin contribution to light baryons in different large-$N$ limits
• Many-body forces with the envelope theory
• Improvement of the envelope theory with the dominantly orbital state method
• An upper bound for asymmetrical spinless Salpeter equations
• The auxiliary field method in quantum mechanics
• Bound cyclic systems with the envelope theory
• Further developments for the auxiliary field method
• Bounds for Hamiltonians with arbitrary kinetic parts
• Upper bounds for critical coupling constants for binding some quantum many-body systems