Arbitrarily Accurate Eigenvalues for General Anharmonic Potentials

Abstract

We show that the Riccati form of the Schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function G. When $G$ and the potential are polynomials, the solutions of these two equations are entire functions (L and K) and the zeroes of K are identical to those of the wave function. Requiring such a zero at a large but finite value of the argument yields the low energy eigenstates with exponentially small errors. Judicious choice of G can improve dramatically the numerical treatment. The method yields many significant digits with modest computer means.

10 pages, 8 figures, uses revtex; new section added, references added

References in corpus (3)

• Higher-Order Corrections to Instantons
• A Simple Method to Make Asymptotic Series of Feynman Diagrams Converge
• Large Field Cutoffs Make Perturbative Series Converge

Cited by in corpus (5)

• Nonlinear Aspects of the Renormalization Group Flows of Dyson's Hierarchical Model
• Quantum anharmonic oscillators: a new approach
• Shifted one-parameter supersymmetric family of quartic asymmetric double-well potentials
• Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators
• A Tractable Example of Perturbation Theory with a Field Cutoff: the Anharmonic Oscillator