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

Abstract

We discuss the relation of a variety of different methods to determine energy levels in lattice QCD simulations: the generalised eigenvalue, the Prony, the generalised pencil of function and the Gardner methods. All three former methods can be understood as special cases of a generalised eigenvalue problem. We show analytically that the leading corrections to an energy $$E_l$$ E l in all three methods due to unresolved states decay asymptotically exponentially like $$\exp (-(E_{n}-E_l)t)$$ 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 $$E_{n}-E_l$$ E n - E l . We illustrate its usage and performance using lattice QCD examples. The Gardner method on the other hand is found less applicable to realistic noisy data.

Mellin moments of spin dependent and independent PDFs of the pion and rho meson

“…refs. [74,[85][86][87][88]) can show that the overlap of smeared vector interpolating currents with the two-pion states is much smaller than for the local currents.…”

Scattering of two and three physical pions at maximal isospin from lattice QCD

“…We refer to Table 9 in the appendix for an overview. We extract the spectrum in each irrep independently using the generalized eigenvalue method (GEVM) [6,88,89] and also the GEVM/PGEVM method [90], see the appendix for more details.…”

“…These principal correlators can be used to build ratios [17,31] or left as-is. All variants can optionally be fed into the Prony generalized Eigenvalue method (PGEVM) [90] with t 0 = 2 fixed to suppress excited states (The PGEVM with δ 0 fixed, see Ref. [90] for details, turned out to not be reliable).…”

Excited states in nucleon structure calculations

“…for a basis of N operators χ = (χ 1 (t), ..., χ N (t)) T with suitable quantum numbers, and solving the generalized eigenvalue problem (GEVP) )) on each timeslice and performing the state assignment going from timeslice t to t + 1 can be a non-trivial task particularly in the presence of an exponentially deteriorating signal-to-noise ratio; for a discussion of methods to sort the states see ref. [101]. The most important feature of the variational approach is that ground state energies and matrix elements are improved with respect to the leading excited state contamination which now depends on the gap E N ( p) − E 0 ( p) to the Nth state in the spectrum [100] instead of the smallest gap in the spectral decomposition of a single two-point function E 1 ( p)−E 0 ( p).…”