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Download PDF by N. N. Bogolyubov: A Method for Studying Model Hamiltonians. A Minimax

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By N. N. Bogolyubov

ISBN-10: 008016742X

ISBN-13: 9780080167428

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Extra resources for A Method for Studying Model Hamiltonians. A Minimax Principle for Problems in Statistical Physics

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2. EQUATIONS OF MOTION AND AUXILIARY OPERATOR INEQUALITIES We turn first of all to the equations of motion for the model Hamil­ tonian Γ. 16) -lGMf)a±f(Jt-C:). -Ca)af. 17) As can be seen, the first two terms in the right-hand sides of these equations originate from the so-called trial Hamiltonian Γα. But this Hamiltonian can be diagonalized by means of a canonical u-v trans­ formation α/=Μ(/)α/-Κ/)αί/ β± / =«(/)α+ / +«*(/)< ν in which we put y/l\ E{f) V2W)| r 2 E(f) = VT*(f)+\Af)\ . As a result, this Hamiltonian is reduced to the form Γα = Σ £ ' ( / ) α / α / + const.

7. 69) in which the operators ßf are arranged in arbitrary order. , fs, some of them can be equal. , fs, we separate out all the distinct indices and denote them by//, .. ,fk, so that each (one or several) of the indices/^, .. , / is equal to one of the indices//, .. ,,f'k. We shall represent the operator product under consideration in the form /*//···&. (1-71) where Bf, denotes a product of operators α^, a^t, all with the same index/'. 73) leaving aside the trivial case when Bf, is identically zero.

OfiO/i . . 63) and note (all the indices^, .. ) that

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A Method for Studying Model Hamiltonians. A Minimax Principle for Problems in Statistical Physics by N. N. Bogolyubov


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