 By Michael Reed, Barry Simon

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This revision will make this e-book extra appealing as a textbook in sensible research. extra refinement of insurance of actual subject matters also will toughen its well-established use as a direction e-book in mathematical physics.

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Additional info for Analysis of Operators

Example text

Starting with Eq. (92), x(t) = x(0) + v(0) 1 (1 − e−βt ) + β β t 1 − eβ(t −t) F(t )dt (118) 0 and multiplying by the force at time t , we have x(t)F(t ) = x(0)F(t ) + v(0) 1 (1 − e−βt )F(t ) + β β t 0 1 − eβ(t −t) F(t )F(t )dt (119) 24 L. Cohen Taking expectation values x(t)F(t ) = 2D β t 1 − eβ(t −t) δ(t − t )dt (120) 0 which evaluates to x(t)F(t ) = 2D β 1 − e−β(t−t ) 0 < t < t 0 otherwise (121) Note that the correlation between position and the force at a later time is zero, x(t)F(t + τ ) = 0 τ >0 (122) The reason for this is that if the force acts at a time later than the position time, it will have had no affect on the position.

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