# 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.

Best differential equations books

Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative (Pure and Applied Mathematics)

Delivering in-depth analyses of present theories and techniques relating to Sobolev-type equations and platforms, this reference is the 1st to introduce a class of equations and structures now not solvable with appreciate to the top order spinoff, and reviews boundary-value difficulties for those periods of equations.

Why the Boundary of a Round Drop Becomes a Curve of Order Four

This e-book issues the matter of evolution of a around oil spot surrounded through water while oil is extracted from a good contained in the spot. It seems that the boundary of the spot continues to be an algebraic curve of measure 4 during evolution. This curve is a dead ringer for an ellipse below a mirrored image with recognize to a circle.

Additional resources for Asymptotic solutions of the one-dimensional Schrödinger equation

Sample text

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.