Parabolic cylinder functions

Near the minimum of the potential x = 0 the wave function is the parabolic cylinder function... 

This equation has the parabolic cylinder functions as standard solutions. Choose the two particular solutions ( ) = U(c,Q and = U o,—Q, which satisfy the desired boundary conditions at a —oo, and b = oo, respectively. 

The gamma function F(c + 1/2) is single valued and analytic over the entire complex c-plane except at c = —n — 1/2 for n = 0,1,2,..., where it has simple poles with residues (-l) /n . The parabolic cylinder functions satisfy... 

This relatively simple governing equation has well understood solutions parabolic cylinder functions. For so-... 

Variational calculus is concerned with finding extrema for example, what is the shortest distance between two points on the surface of a parabolic cylinder In ordinary, garden-variety calculus, we deal wiihfunctions, which are objects whose values depend on the values of numerical quantities. But in the variational calculus, the focus of attention is on functionals, which are objects whose values depend on functions. For example, we may interpret the entropy as a functional because its value depends on other thermodynamic functions, such as temperature, pressure, and composition. Since the functionals differ from functions, we sometimes find it convenient to use a notation for operators on functionals that differs somewhat from the notation for operators on functions. For our purposes, the most important notational distinction occurs for differential operators. 

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