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Unformatted text preview: Physics 498/MMA Handout 4 Fall 2007 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ ∼ mstone5 Prof. M. Stone 2117 ESB University of Illinois 1) Linear differential operators : a) Consider the differential operator ˆ L = id/dx . Find the formal adjoint of L with respect to the inner product h u  v i = R wu * v dx , and find the corresponding surface term Q [ u, v ]. b) Now do the same for the operator M = d 4 /dx 4 , for the case w = 1. Find the adjoint boundary conditions defining the domain of M † for the case D ( M ) = { y, y (4) ∈ L 2 [0 , 1] : y (0) = y 000 (0) = y (1) = y 000 (1) = 0 } . (Hint: you will find an identity from homework 2 to be very useful.) 2) SturmLiouville forms : By constructing appropriate weight functions convert the fol lowing common operators into SturmLiouville form: a) ˆ L = (1 x 2 ) d 2 /dx 2 + [( μ ν ) ( μ + ν + 2) x ] d/dx. b) ˆ L = (1 x 2 ) d 2 /dx 2 3 x d/dx. c) ˆ L = d 2 /dx 2 2 x (1 x 2 ) 1 d/dx m 2 (1 x 2 ) 1 ....
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 Fall '07
 Stone
 Derivative, Eigenfunction, Differential operator, Boundary conditions, Prof. M. Stone, ESB University of Illinois

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