It is known that the exact eigen-functions of sturm-liouville problem are orthogonal with respect to the weight function. But what happens when the solutions are obtained using approximate analytical method?
Do the approximate analytical solutions obtained using asymptotic techniques or using WKB method orthogonal?
Logically, If the solutions are the same, orthogonality property would still be valid irrespective of the methods that yielded the solution. However, you can subject each of the equivalent solutions to orthogonality test to be certain.
The answer is that approximate solutions will be approximately orthogonal - but this is a vacuous statement unless 'approximately' is quantified. In practice, many analytical approximation techniques will yield not 'a' solution but a *series* of solutions approximating the actual - unavailable - one with increasing accuracy. (What 'accuracy' means is another story.) WKB, for example, will yield an asymptotic expansion of the actual solution, and this expansion can be truncated at different orders to yield different approximations. (Often, one retains only the principal part/leading order term, but that depends strongly on what one needs to do.) That having been said, truncations of two distinct eigenfunctions will typically *not* be mutually orthogonal - you can check that this is the case with the leading order terms of your favorite eigenfunction set. One could argue that they deviate from orthogonality by an amount (angle) of the order of the small parameter used to expand, but in reality it's impossible to say much more without knowing the specific nature of the problem in question.
Thanks Zagaris. I agree that after truncation, the two distinct eigenfunctions typically won't be orthogonal. However you said, they might deviate from orthogonality by the order of parameter used to expand. How is this deviation measured? Is it the deviation from the value zero?
You can measure the angle formed by two functions using the L^2 inner product associated with your Sturm-Liouville problem: cos(phi) = / ( ||f||*||g|| ) . In my experience with WKB, computing the corresponding integrals may be impossible and further approximations (e.g. the Laplace approximation) might need to be made.
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