Download Asymptotic solutions of the one-dimensional Schrödinger by S. Yu. Slavyanov PDF

By S. Yu. Slavyanov

This ebook is dedicated to asymptotic research of strategies of moment order traditional differential equations with a small parameter. the most emphasis is on numerous optimistic schemes of acquiring asymptotic suggestions, their merits and disadvantages, and particular computations. the writer supplies a whole evaluate of the nation of the idea and in addition concentrates on a few lesser identified points and difficulties, specifically the issues during which exponentially small phrases could be taken under consideration or the research of equations with shut transition issues. Such functions because the derivation of the formulation for the quasiclassical quantizations, spectrum splitting in a symmetrical strength, etc., are thought of.

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13). 3. The Whittaker functions and their asymptotics. The confluent hypergeometric equation contains the term with the first derivative. The description of asymptotic solutions of second order equations becomes more simple if this term vanishes. 1), and introduce the parametersµ and x: µ = (c -1)/2, x= c/2-a. 19) 1 1/4 -µ 2 ) y(z)=O, y"(z)+ ( --+~+ 4 z z2 which does not contain the first derivative. 21) Y1{z) = Mx,µ(z) = e-z 2 l 2zµ+lq>(µ- X + 1/2,2µ+ 1,z), Y2{z) = Wx,µ(z) = e-z 2 / 2zµ+i/ 2w(µ - x + 1/2, 2µ + 1, z).

19) 1 1/4 -µ 2 ) y(z)=O, y"(z)+ ( --+~+ 4 z z2 which does not contain the first derivative. 21) Y1{z) = Mx,µ(z) = e-z 2 l 2zµ+lq>(µ- X + 1/2,2µ+ 1,z), Y2{z) = Wx,µ(z) = e-z 2 / 2zµ+i/ 2w(µ - x + 1/2, 2µ + 1, z). 19). The Stokes lines are the positive and negative real axes. ,,. r(µ- x + 1/2) . r(-µ- x + 1/2) z , k=l 92(µ,x,z) =91(µ,-x,z). I. µ (z ) -_ e-z/2 z X Y1 ( x, µ, z ) , Iargzl :::; 11' - e. 18): r(2µ + 1) e-z/2zxei1r(µ-x+1/2)g (µ x z) 1 r(µ + x + 1/2) ' ' + r(2µ + 1) ezl2z-(µ+x+1/2)g (µ x z) 2 ' ' ' r(µ - x + 1/2) e:::; Iargzl:::; 11' - e, r(2µ + 1) e-z/2zXe-i1r(µ-x+1/2)g (µ X z) 1 r(µ + x + 1/2) ' ' + r(2µ + 1) ez/2z-(µ+x+1/2)g (µ x z) 2 r(µ - x + 1/2) ' ' ' -11' + e:::; Iargzl:::; -€, r(2µ + 1) -x/2 x [ Mx,µ (x ) -r(µ + x + l/ 2) e x cos 11'(µ- x + 1/2)]g1(µ,x,x) + Mx,µ(-x) r(2µ + 1) exl2x-(µ+x+1/2)g (µ x x) 2 ' ' r(µ - x + 1/2) ' r(2µ + 1) x/2 x r(µ + x + 1/2) e x g1(µ, x.

3. PARABOLIC CYLINDER FUNCTIONS AND THEIR ASYMPTOTICS 29 e 3. The following notations are used: ( = z 2 /4, = x 2 /4, ao = 1, bo = 1, bk(11) = ak(-11-l), ak+i(11) = (k2" + v(~k 1 >)ak(11) r(2k-v) or ak (11 ) = k! 2a,. r(-v). /2ir 2-v+l(-(v+l)/2e< r(-11) x ( 1+ D,,(z) 2"ei11'v/2(v/2ef. f2ir 2-v-le-(v+l)/2e-F. 2r(-11) x (1 + 2" COS11"11e-F. ( 1 + z=-x f ~(-1)kbk(11)e-k) (-l)kak(11)e-k) k=l + rY::) 2-v-lef. ( 1 + ~bk(11)e-k) We see that on the positive real axis the asymptotics of the function V,,(x) contains only the dominant part.

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