A Method for Studying Model Hamiltonians. A Minimax by N. N. Bogolyubov

By N. N. Bogolyubov

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2 ί ¥ £ />(/)} ^ w f e IGJ (^-ca)(y+-ca*)+^J. | ( / . - C J ( 7 + - C Ö + - ^ j . 28) We also obtain, entirely analogously, RfR} ^ 2 Ρ ( / ) | Σ I Gx\{J«-CMJt-Ct)+~>\. 30) § 3. 29), this problem reduces to the problem of obtaining bounds for D. ßX)r. 38 A METHOD FOR STUDYING MODEL HAMILTONIANS We shall move the operator (Ja—CJ successively to the left. We find D. -CJßA . . ßf,{Ji-Cl)ßZ ... ßj-)r, °'32) 7= 1 where the operator Bj is obtained if the operator ßf in the product ßh ■ ■ ■ ßf. 34) To calculate this difference, we shall examine two separate cases: We have α//α-Λα^ = ßf, = °% and ßf, = «ft· ^YtKWfafflfatf-afatf*/} = 2V Σ W){[«//*/ +β/a/Jöiz-a/Iaz/iiz + ei/a/J}.

61), will be equal to zero if they are calculated on the basis of the trial Hamilto­ nian Γα. 62) where the averages are of the products treated earlier. Thus, of all the products ßh · · · ßf. e. with all creation operators α^" to the left and all annihilation operators onf to the right) we shall only have to consider products «Λ · · · αΛαΛ' · · · <*/*' in which each of the indices fv . . , fk is equal to one of the indices / / , .. ,Λ'. B u t s u c h products can obviously be reduced to the form We put ±«Λ · · · αΛαΛ · · · α/*· (Vk = «£ .

29), this problem reduces to the problem of obtaining bounds for D. ßX)r. 38 A METHOD FOR STUDYING MODEL HAMILTONIANS We shall move the operator (Ja—CJ successively to the left. We find D. -CJßA . . ßf,{Ji-Cl)ßZ ... ßj-)r, °'32) 7= 1 where the operator Bj is obtained if the operator ßf in the product ßh ■ ■ ■ ßf. 34) To calculate this difference, we shall examine two separate cases: We have α//α-Λα^ = ßf, = °% and ßf, = «ft· ^YtKWfafflfatf-afatf*/} = 2V Σ W){[«//*/ +β/a/Jöiz-a/Iaz/iiz + ei/a/J}.

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