Ground state energies from converging and diverging power series expansions

Abstract

It is often assumed that bound states of quantum mechanical systems are intrinsically non-perturbative in nature and therefore any power series expansion methods should be inapplicable to predict the energies for attractive potentials. However, if the spatial domain of the Schrödinger Hamiltonian for attractive one-dimensional potentials is confined to a finite length L, the usual Rayleigh-Schrödinger perturbation theory can converge rapidly and is perfectly accurate in the weak-binding region where the ground state's spatial extension is larger than L. Once the binding strength is so strong that the ground state's extension is less than L, the power expansion becomes divergent, consistent with the expectation that bound states are non-perturbative. However, we propose a new truncated Borel-like summation technique that can recover the correct bound state energy from the diverging sum. We also show that perturbation theory becomes divergent in the vicinity of an avoided-level crossing. Here the same numerical summation technique can be applied to reproduce the correct finite energies from the diverging perturbative sums.

Comment on: “Ground state energies from converging and diverging power series expansions”, Ann. Phys. 373 (2016) 456–469

“…that agrees with the general expression (5) through the fourth term. If we only keep the first term in the right-hand side we obtain E ≈ −2mV 2 0 /(h 2 b 2 ) that agrees with the result of Lisowski et al [1]…”

“…when β → 0. The next example is the Dirac-delta-potential v(x) = −δ(x) already studied by Lisowski et al [1]. Upon choosing the exact boundary conditions lim |x|→∞ ψ(x) = 0 we obtain the dimensionless energy ϵ = −λ 2 /4 for the only bound-state eigenvalue.…”

“…It is worth noting that if instead of periodic boundary conditions we impose Neumann ones ψ ′ (±L/2) = 0 then we obtain exactly the same quantization condition (18). Finally we consider the exponential potential V (x) = −V 0 e −b|x| also discussed by Lisowski et al [1]. Upon choosing the length γ = 1/b we obtain v(x) = −e |x| , λ = 2mV 0 /(h 2 b 2 ) and ϵ = 2mE/(h 2 b 2 ).…”

“…It is well known that there exists a perturbation series about V 0 = 0 in the case of a onedimensional short-range potential and there are even explicit expressions for the first perturbation coefficients [2] (and references therein). However, Lisowski et al [1] state that ''It is often assumed that bound states of quantum mechanical systems are intrinsically non-perturbative in nature and therefore any power series expansion methods should be inapplicable to predict the energies for attractive potentials''. The purpose of this paper is to discuss the connection between the approximate perturbation series proposed by Lisowski et al [1] and the well known exact perturbation series [2].…”

Space-time-resolved Breit-Wheeler process for a model system

“…We should point out that despite the approximations concerning the spatial dimensions, the fermions' spin directions, and non-vanishing mass of the model photons, due to its fundamental character this Hamiltonian (and its reduced versions) has been studied in a wide variety of contexts. For example, to name a few, this includes the examination of the origin of attractive and repulsive interfermionic forces due to the exchange of intermediating photons [23], the evolution of bare electrons into physical electrons under photon emission [24,25], a space-time resolved Compton scattering [26], the formation of fermion bound states in the perturbative and also Borel summable non-perturbative regimes [22,27,28].…”