On the generalised eigenvalue method and its relation to Prony and generalised pencil of function methods

Abstract

We discuss the relation of three methods to determine energy levels in lattice QCD simulations: the generalised eigenvalue, the Prony and the generalised pencil of function methods. All three can be understood as special cases of a generalised eigenvalue problem. We show analytically that the leading corrections to an energy $E_l$ in all three methods due to unresolved states decay asymptotically exponentially like $\exp (-(E_n-E_l)t)$. Using synthetic data we show that these corrections behave as expected also in practice. We propose a novel combination of the generalised eigenvalue and the Prony method, denoted as GEVM/PGEVM, which helps to increase the energy gap $E_n-E_l$. We illustrate its usage and performance using lattice QCD examples.

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• Three-body resonances in the ${\phi }^4$ theory
• Visualizing resonances in finite volume
• Mellin moments of spin dependent and independent PDFs of the pion and rho meson
• Towards studying the structural differences between the pion and its radial excitation
• Automated label flows for excited states of correlation functions in lattice gauge theory