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Extremum principles for the equation nabla-squared phi = phi - phi-to-the-third

Authors
  • Robinson, P.D.
Publication Date
Jan 01, 1971
Identifiers
DOI: 10.1063/1.1665478
OAI: oai:inspirehep.net:71296
Source
INSPIRE-HEP
Keywords
License
Unknown
External links

Abstract

It has been conjectured that the Lagrangian functional G(Φ)=∫ [ 1 2 Φ(−∇ 2 +1)Φ− 1 4 Φ 4 ] d r provides an upper bound for G(φ0), where φ0 is the ground‐state eigenfunction of the nonlinear field equation −∇2φ + φ − φ3 = 0, provided that G(Φ) is constrained to be stationary with respect to variations in the amplitude of Φ. In this paper we demonstrate that this conjecture is true. The effect of a stationary‐scale constraint on G(Φ) is also shown to guarantee an upper bound. Complementary functionals arising from an Euler‐Hamilton approach to the problem are investigated. Unfortunately, these do not (as in more favorable circumstances) provide lower bounds for G(φ0), but merely alternative upper bounds. With a very simple trial function, a complementary bound is closer to G(φ0) than is G(Φ).

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