Beta-Kernel Truncation Bases for the Functional Renormalization Group: Complete Error Analysis and Adaptive Nyström Propagator Approximation
Authors/Creators
Description
We construct a family of Beta-kernel basis functions u_m^(p)(t) = C_{m,p} t^m(1−t)^p for truncating the effective action and performing Nyström approximation of the regularised propagator in the Functional Renormalization Group (FRG). Working in one-dimensional quantum mechanics with a mass regulator R_k = k², we derive the closed-form Green's function and establish three main results: a complete integration-by-parts error bound for the monomial truncation basis yielding cumulative truncation error E_N = O(N⁻¹); a rank-one collapse theorem showing the regularised propagator reduces to a rank-one operator in the large-m limit; and a closed ODE system for propagator values and adaptive Nyström nodes via the Dirac–Frenkel–McLachlan variational principle, with exact self-consistency for a single node at the midpoint. All analytical estimates are confirmed by explicit numerical computations at M_k = 1.
Files
BKTB.pdf
Files
(550.4 kB)
| Name | Size | Download all |
|---|---|---|
|
md5:1c1927cf4dc78d897b56b33d79faa72a
|
550.4 kB | Preview Download |
Additional details
Dates
- Created
-
2026-05-14
References
- [1] C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90–94. DOI: 10.1016/0370-2693(93)90726-X. [2] J. Berges, N. Tetradis, C. Wetterich, Non-perturbative renormalization flow in quan- tum field theory and statistical physics, Phys. Rep. 363 (2002) 223–386. DOI: 10.1016/S0370-1573(01)00098-9. [3] C. Bagnuls, C. Bervillier, Exact renormalization group equations: an introductory review, Phys. Rep. 348 (2001) 91–157. DOI: 10.1016/S0370-1573(00)00137-X. [4] B. Delamotte, An introduction to nonperturbative renormalization, Lect. Notes Phys. 852 (2012) 49–132. DOI: 10.1007/978-3-642-27320-9_2. [5] N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, N. Wsche- bor, The nonperturbative functional renormalization group and its applications, Phys. Rep. 910 (2021) 1–114. DOI: 10.1016/j.physrep.2021.01.001. [6] P. Billingsley, Convergence of Probability Measures, 2nd edn, Wiley, New York, 1999. ISBN: 978-0-471-19745-4; DOI: 10.1002/9780470316962. [7] R. J. Serfling, Approximation Theorems of Mathematical Statistics, Wiley, New York, 1980. ISBN: 978-0-471-02403-3; DOI: 10.1002/9780470316481. [8] H. A. David, H. N. Nagaraja, Order Statistics, 3rd edn, Wiley, Hoboken, 2003. ISBN: 978-0-471-38926-2. [9] J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, Oxford, 2005. ISBN: 978-0-19-856674-8. DOI: 10.1093/acprof:oso/9780198509233.003.0002. [10] M. V. Berry, K. E. Mount, Semiclassical approximations in wave mechanics, Rep. Prog. Phys. 35 (1972) 315–397. DOI: 10.1088/0034-4885/35/1/306. [11] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New York, 1990. ISBN: 0-387-97173-4. [12] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw- Hill, New York, 1955. ISBN: 978-0-07-099256-6. [13] J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, 1978. ISBN: 978-0-89874-957-1. [14] M. H. Beck, A. Jäckle, G. A. Worth, H.-D. Meyer, The multiconfiguration time-dependent Hartree method: a highly efficient algorithm for propagating wavepackets, Phys. Rep. 324 (2000) 1–105. DOI: 10.1016/S0370-1573(99)00047-2. [15] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Nu- merical Analysis, European Mathematical Society, Zürich, 2008. DOI: 10.4171/067.