By Barus C.
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Extra info for The Open Mercury Manometer Read by Displacement Interferometry
Sample text
Instead of increasing L and N independently of one another, we coupled them such that N ˙ L2 . In this way the largest energy that can be resolved by the basis set, Emax ˙ N 2 =L2 (the i are particle-in-a-box eigenfunctions), and the smallest energy, Emin ˙ 1=L2 , improve by the same ratio as L is increased. Note that Emax would have remained constant if N had been chosen proportional to L. For the ÿrst three resonances, we have determined as a function of basis size the point Áopt of optimal stabilization, the complex energy E|Áopt at that point, as well as the degree of stabilization |Á dE=dÁ|Áopt .
S. Cederbaum / Physics Reports 368 (2002) 1 – 117 51 it is tempting to associate limj→0+ Uˆ j (0; −∞; Á)| 0N ) with | 0N (Á)). However, from our experience with Eqs. (35) – (37) we expect that limj→0+ Uˆ j (0; −∞; Á)| 0N ) is not well deÿned. A suitable quotient must be formed. According to Gell-Mann and Low [99], if the quantity Uˆ j (0; −∞; Á = 0)| 0N ) (186) lim+ N N ˆ j→0 ( 0 |U j (0; −∞; Á = 0)| 0 ) exists to all orders of perturbation theory, then it is an eigenstate of Hˆ (Á = 0). , Á → 0+ .
47]), than to derive each term by purely algebraic manipulation of Eq. (188). A ÿrst important application of the diagrammatic analysis is the linked-cluster theorem [103,104]: The numerator in Eq. (188) factorizes into an expression identical with the denominator—which can therefore be cancelled—and a factor that can be expanded in a series of the so-called connected diagrams. ) = −i = pp ∞ −∞ d(t − t )( ˆ ˆ N ˆ iHˆ 0 t cp e− i H 0 t ei H 0 t 0 |T [e np 1 − np + ! − p + ij ! (t −t ) 0 )e (192) 52 R.