Do evanescent waves span a vector space?

[Heirot]

In dealing with Poisson, Laplace, Schrodinger and other wave equations, one has to deal with propagating and evanescent waves. We know all about the propagating waves - orthogonality and completeness relations, but what about evanescent waves? Do they form a vector space with corresponding identities?

 

[Eynstone]

I wonder how evanescent waves could satisfy the completeness condition. It seems quite unlikely.

 

[arkajad]

That depends on your precise definition of evanescent waves, then on the appropriate definition of the scalar product, which will be rather a different one than the usual one. You will get a pre-Hilbert space and, after the completion - a Hilbert space. Because of the different scalar product - the identities will be different.

 

[Heirot]

By evanescent waves, I mean Exp[-k x] with both k and x real.

 

[arkajad]

Just "real", or "positive"?

 

[Heirot]

For now, I'm going to go with "real". I want to allow all possibilities.

 

[arkajad]

Are you in a finite box or in an infinite space? When you say "othogonality" relations for waves, what do you mean by these. Waves are not square integrable in infinite space.

 

[Heirot]

I have a problem with the lack of symmetry between propagating and evanescent waves. Propagating waves can be treated in infinite space in the distributional sense, so that they span the "plane wave" basis. Why can't I expand any function on [0, oo> that goes to zero at oo in the evanescent wave basis?