Numerical approaches of field cutoff in quantum mechanics

• Our idea of field cutoff was initiated a few years ago by numerical calculations in quantum mechanics.
• We have refined this approach showing that the Riccati form of the one-dimensional Schrödinger equation can be reformulated in terms of two first order linear equations depending on an arbitrary function G.
• When G and the potential (as for anharmonic oscillators) 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.
• Approximate formulas for these errors were provided

   
  Figure 6: Various analytical error formulas explained in the article versus the field cutoff.
  

• The arbitrary function G can be chosen in order improve dramatically the numerical treatment.
• More details in: Y. Meurice, Arbitrarily Accurate Eigenvalues for One-dimensional Polynomial Potentials, J. Phys. A 35, 8831 (2002).

Work in progress, plans

• We have found a generalization of the method for problems involving several oscillators.
• We are planning to apply it to the problem with . In two limits ( where the two Fourier modes decouple and where ) we have an additional constant of motion.
• By increasing the energy, we can observe bifurcations in the stability of periodic solutions illustrated below.
• We plan to study this classical instability in terms of small denominators appearing in the perturbative series. We plan to study its quantum counterpart (degeneracies) perturbatively and confront this study with numerical results.

 
Figure 7: Poincaré sections for E=0.3055

 
Figure 8: Poincaré sections for E=0.34