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Perturbation Theory Based on Exactly Localized States.

Cecil Rousseau

Published Jan 1, 1968• 0 citations
OpenAlex

Abstract

We investigate the usefulness of a perturbation theory which starts from exactly localized states. We find that for various single particle problems, the ground-state energy as a function of the effective coupling strength is a series of Stieltjes and therefore, in principle, summable by means of the Pade� approximant. For several particle problems the Pade� approximant is inadequate and we have introduced a generalized approximant which is very effective if the number of interacting particles is not too great.

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We investigate the usefulness of a perturbation theory which starts from exactly localized states. We find that for various single particle problems, the ground-state energy as a function of the effective coupling strength is a series of Stieltjes and therefore, in principle, summable by means of the Pade� approximant. For several particle problems the Pade� approximant is inadequate and we have introduced a generalized approximant which is very effective if the number of interacting particles is not too great.
We present a new formulation of perturbation theory for quantum systems, designated here as: “mean field perturbation theory” (MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of arbitrary strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the “standard formulation of perturbation theory” (SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the nonlinearity and the analytic structure (in the coupling strength) of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since “renormalon”-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case.
We introduce a non perturbative general approximation scheme (NGAS) that can handle interactions of any strength in quantum theory. This approach starts with an input Hamiltonian that can be solved exactly. The interaction effects are then built into this Hamiltonian through nonlinear feedback enforced by self consistency conditions. While the method itself is nonperturbative it can be systematically improved using a new perturbation method called 'mean field perturbation theory' which does not involve power series expansion in any small parameter. We put this scheme to the test on one dimensional anharmonic interactions using the harmonic approximation. The results are consistently accurate across various cases including quartic, sextic, and octic anharmonic oscillators, as well as the quartic double well oscillator (QDWO) even when the coupling strength varies widely. The flexibility of the method is demonstrated when we swap the input Hamiltonian for that of an infinite square well and still achieve comparable accuracy. When applied to the λϕ4 quantum field theory this approach aligns with the Gaussian effective potential method under the harmonic approximation. Beyond that it reveals the condensate structure of the effective vacuum and highlights the instability of the perturbative ground state. Notably, our ground-state energy results for the QDWO stand in stark contrast to those from standard perturbation theory where Borel summation fails regardless of coupling strength.
We generalize a recently developed perturbation approach [F. M. Fern\'andez and E. A. Castro, Phys. Rev. A 46, 7288 (1992)], extending its applicability to a wider variety of separable quantum-mechanical problems. As an illustrative example, we consider a particle in a one-dimensional box with a finite potential wall perturbed by a linear potential and obtain the perturbation corrections to the wave function and energy of an arbitrary state in terms of the corresponding unperturbed quantities.
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.
A precise calculation of the ground-state energy of the complex 𝒫𝒯-symmetric Hamiltonian H=p2+14x2+iλx3, is performed using high-order Rayleigh–Schrödinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. Here we present convincing numerical evidence that the Rayleigh–Schrödinger perturbation series is Borel summable, and show that Padé summation provides excellent agreement with the real energy spectrum. Padé analysis provides strong numerical evidence that the once-subtracted ground-state energy considered as a function of λ2 is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian H=p2+14 x2−εx3.