The Schrödinger equation for quantum fields with nonlinear nonlocal scattering

James Glimm

doi:10.1007/BF01773357

The Schrödinger equation for quantum fields with nonlinear nonlocal scattering

James Glimm

Applied Mathematics and Statistics

Stony Brook University

Research output: Contribution to journal › Article › peer-review

Abstract

This paper considers perturbations H=H₀+εV of the Hamiltonian operator H₀ of a free scalar Boson field. V is a polynomial in the annihilation creation operators. Terms of any order are allowed in V, but point interactions, such as ∫:0(x)⁴(x)⁴:dx, are not considered. Unnormalized solutions for the Schrödinger equation are found. For ε→0, these solutions have a partial asymptotic expansion in powers of ε. The set of all possible pertubation terms V forms a Lie algebra. General properties of this Lie algebra are investigated.

Original language	English
Pages (from-to)	271-300
Number of pages	30
Journal	Communications in Mathematical Physics
Volume	2
Issue number	1
DOIs	https://doi.org/10.1007/BF01773357
State	Published - Dec 1966

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title = "The Schr{\"o}dinger equation for quantum fields with nonlinear nonlocal scattering",

abstract = "This paper considers perturbations H=H0+εV of the Hamiltonian operator H0 of a free scalar Boson field. V is a polynomial in the annihilation creation operators. Terms of any order are allowed in V, but point interactions, such as ∫:0(x)4(x)4:dx, are not considered. Unnormalized solutions for the Schr{\"o}dinger equation are found. For ε→0, these solutions have a partial asymptotic expansion in powers of ε. The set of all possible pertubation terms V forms a Lie algebra. General properties of this Lie algebra are investigated.",

author = "James Glimm",

year = "1966",

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N2 - This paper considers perturbations H=H0+εV of the Hamiltonian operator H0 of a free scalar Boson field. V is a polynomial in the annihilation creation operators. Terms of any order are allowed in V, but point interactions, such as ∫:0(x)4(x)4:dx, are not considered. Unnormalized solutions for the Schrödinger equation are found. For ε→0, these solutions have a partial asymptotic expansion in powers of ε. The set of all possible pertubation terms V forms a Lie algebra. General properties of this Lie algebra are investigated.

AB - This paper considers perturbations H=H0+εV of the Hamiltonian operator H0 of a free scalar Boson field. V is a polynomial in the annihilation creation operators. Terms of any order are allowed in V, but point interactions, such as ∫:0(x)4(x)4:dx, are not considered. Unnormalized solutions for the Schrödinger equation are found. For ε→0, these solutions have a partial asymptotic expansion in powers of ε. The set of all possible pertubation terms V forms a Lie algebra. General properties of this Lie algebra are investigated.

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DO - 10.1007/BF01773357

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SP - 271

EP - 300

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -

The Schrödinger equation for quantum fields with nonlinear nonlocal scattering

Abstract

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