Effect of the boundary conditions on the accuracy of perturbation theory

1987 ◽  
Vol 55 (10) ◽  
pp. 924-929
Author(s):  
M. Coronado ◽  
N. Domínguez ◽  
J. Flores ◽  
C. de la Portilla
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

scholarly journals A note on boundary conditions in Euclidean gravity

2021 ◽  
pp. 2140004
Author(s):  
Edward Witten
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Perturbation Theory ◽  
Quantum Gravity ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Conformal Geometry ◽  
Extrinsic Curvature ◽  
Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

scholarly journals ON DUALITY AND REFLECTION FACTORS FOR THE SINH–GORDON MODEL WITH A BOUNDARY

1998 ◽  
Vol 13 (16) ◽  
pp. 2709-2722 ◽  
Author(s):  
E. CORRIGAN
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Semiclassical Approximation ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Free Fermion ◽  
Different Boundary Conditions ◽  
Boundary Potential ◽  
Dual Relationship ◽  
Integrable Boundary Conditions

The sinh–Gordon model with integrable boundary conditions is considered in low order perturbation theory. It is pointed out that results obtained by Ghoshal for the sine–Gordon breather reflection factors suggest an interesting dual relationship between models with different boundary conditions. Ghoshal's formula for the lightest breather is checked perturbatively to O(β2) in the special set of cases in which the ϕ→-ϕ symmetry is maintained. It is noted that the parametrization of the boundary potential which is natural for the semiclassical approximation also provides a good parametrizaiton at the "free-fermion" point.

scholarly journals Perturbation theory in quantum mechanics—II

1929 ◽  
Vol 124 (793) ◽  
pp. 176-188 ◽  
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Stark Effect ◽  
Initial Conditions ◽  
Formal Solution ◽  
Fixed Time ◽  
Characteristic Functions ◽  
Perturbation Equation ◽  
Experimental Conditions ◽  
General Method

In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H 0 , H 1 , H 2 ( t ), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H 0 + H 2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H 0 + H 2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H 0 , H 1 , H 2 ( t ) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H 0 , H 1 , H 2 ( t ). To obtain Born’s theory we take H 1 = 0, and for Oppenheimer’s we take H 2 ( t ) = H 2 (T) and T = 0. In Dirac’s theory we have H 1 = 0, H 2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.

scholarly journals Volume independence for Yang–Mills fields on the twisted torus

2014 ◽  
Vol 29 (25) ◽  
pp. 1445001 ◽  
Author(s):  
Margarita García Pérez ◽  
Antonio González-Arroyo ◽  
Masanori Okawa
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Gauge Theories ◽  
Field Theories ◽  
Gauge Fields ◽  
Dimensional Torus ◽  
Loop Order ◽  
Yang Mills ◽  
Hooft Coupling ◽  
Twisted Boundary Conditions

We review some recent results related to the notion of volume independence in SU (N) Yang–Mills theories. The topic is discussed in the context of gauge theories living on a d-dimensional torus with twisted boundary conditions. After a brief introduction reviewing the formalism for introducing gauge fields on a torus, we discuss how volume independence arises in perturbation theory. We show how, for appropriately chosen twist tensors, perturbative results to all orders in the 't Hooft coupling depend on a specific combination of the rank of the gauge group (N) and the periods of the torus (l), given by lN2/d, for d even. We discuss the well-known relation to noncommutative field theories and address certain threats to volume independence associated to the occurrence of tachyonic instabilities at one-loop order. We end by presenting some numerical results in 2+1 dimensions that extend these ideas to the nonperturbative domain.

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