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N. N. Bogolyubov's A Method for Studying Model Hamiltonians. A Minimax PDF

By N. N. Bogolyubov

ISBN-10: 008016742X

ISBN-13: 9780080167428

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59) k + q even. We now take the opposite case. Let C/i = α^ . . OfcOfl . . a/,', q > fc Ξ>0; fc+# even. We have C/ί = ayt . . α£αΛ . . 59). But |r| = |<ϋ>ί| = |r|, and therefore \(ύ ... κ + α / .... ) Γ | ^ ^ ]fsv+^-. 60) even. 9fkin this product there is at least one index ft which is not equal to any of the indices//, . . 58) it follows again that I (4 ... fa ... «β>Γ| ^ x ] / ' ^ + ^ a·«) 52 A METHOD FOR STUDYING MODEL HAMILTONIANS (if any one of the indices^, .. 61), will be equal to zero if they are calculated on the basis of the trial Hamilto­ nian Γα.

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

With the intention of using the situation when af and af are found to be "squeezed" between (Ja—Ca) and (/+ — C*), we shall have to interchange these factors, since aj commutes with Ja and af with J*. -JJt = -— ^ / ^ ( α ^ α ί , - α ί , α , , + ^ - α / α , ) = 2^^:ι^σ)ΐ2(«/«/-«/»^2^-Σΐ^(/)ΐ2· Hence follows =s a/(/ a -C a ) ( / + - O V + ^ £ I W ) I2 = (/ a -C a )a/a,(7+-C* a ) + ^ £ \XJJ)\* ^ ( / . - C J ( / + - C : ) + ~ - £ |-U/)I 2 . *) + 2 ί ¥ £ />(/)} ^ w f e IGJ (^-ca)(y+-ca*)+^J. | ( / .

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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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