Action for a discrete potential

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Hello,

I have a potential function given numerically at points evenly spaced. That is to say, I have the numerical values of [tex]V(0), V(\delta), V(2\delta), V(3\delta), ...[/tex], in some interval. I need to calculate the action integral in terms of initial and end-points: [tex]S(x_b, t_b; x_a, t_a)[/tex].

I think I first need an analytical expression for potential to start with. Since I only know potential for some few points, I tried to write down an interpolation function. However, there's one restriction: the potential function can be quadratic in [tex]x[/tex] at most. Quadratic spline interpolation gives just that sort of function, but the problem with it is, the method gives a piecewise function for potential. One function defined within [tex]x \textrm{ within } [0, \delta][/tex], another for [tex]x \textrm{ within } [\delta, 2\delta][/tex], and so on. The potential (and it's derivative) generated with quadratic spline is continuous, but well, it's piecewise.

So, I wonder:
  • Is there a way for calculating [tex]S(x_b, t_b; x_a, t_a)[/tex], using a piecewise function?
  • Or is there a way to write down an interpolation function that is quadratic,and not piecewise?
  • Or any way around without ever writing down an interpolation?

Any help or advice will be appreciated.
Thanks.


Note: The problem is actually quantum-mechanical, and the restriction to quadratic functions stems from the fact that, when the potential does not contain 3rd or higher orders, one can use [tex]K(x_b, t_b; x_a, t_a) = F(t_b,t_a) exp{(i/\hslash)S(x_b, t_b; x_a, t_a)}[/tex].
 
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