Numerical Strategies for Solving Multiparameter Spectral Problems

Considerations about the incompleteness of the Ehrenfest’s theorem in quantum mechanics

“…Equation (41) shows flagrantly and unequivocally the evidence that the answer to the question we asked at the end of section 2 regarding the sufficiency of the Hamiltonian's hermiticity to imply the unconditional vanishing of the integral I Ω should be negative. Indeed, equation ( 29) tells us that the Hamiltonian is hermitean regardless of which wavefunction boundary conditions are enforced but equation (41) clearly says that the integral I Ω=x does not vanish with the periodic and vanishing-derivative boundary conditions. The substitution of the position operator Ω = x in equation ( 39) leads to…”

“…Here again we see that the integral may differ from zero according to the enforced boundary conditions; concerning confinement and periodicity, the situation in equation ( 49) is reversed with respect to equation (41). The substitution of the momentum operator Ω = p in equation ( 39) leads to the well known force term…”

“…and, therefore, we concentrate only on the solutions with positive α; they describe the situations in which electric charge and electric field have same sign. For the numerical solution of our mathematical problem, we have utilised a method based on high-order finite differences [39][40][41] implemented in the code HOFiD_MSP that can solve multiparameter spectral BV-ODE problems; in our calculations we have always utilised 8th-order formulae on a grid composed of 1001 equally spaced points on the interval [−1, +1].…”

BVPs Codes for Solving Optimal Control Problems

“…The strategy adopted returns a code suitable to solve high-order boundary value problems that can be singularly perturbed, singular, with discontinuous terms and multipoint. Other versions of the code solve singular second order initial value problems [38], Sturm-Liouville problems [39] and multi-parameters spectral problems [40].…”