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My question is somewhat broad, but I do not know how to precisely state the issue. I am investigating stability of certain class of scalar PDE on $\mathbb{R}$.

Previous work in this topic has introduced exponentially weighted space to look into this, e.g.

$H_a=L^2[\mathbb{R},e^{-x^2}dx]$.

I am trying to understand the effect of introducing such spaces on the discrete and continuous spectrum of a Linear operator (coming from linearizing the said PDE).

1). For example, does the point spectra remain same ?

Are there any references (papers/books/monographs) which discuss this issue ?

2). A related question is this. If we work in $H_a$, a natural way to tackle is introduction of Hermite basis, which is countable.

If we work in $H_b=L^2[\mathbb{R},dx]$, one would use Fourier transform and have an uncountable number of basis functions. Does this difference in number of basis relate to 1).

To give a specific example, one can consider the following operator $L$, where

$Lu(x)=[icx+\partial_{xx}]u(x)$.

If one is trying to find the essential spectrum of this operator acting on $H_b$, then the usual technique of looking at dispersion relation of asymptotic operator (i.e. $x\rightarrow \pm\infty$) fails since the first term in the operator blows up.

So there is hope (?) that if we work in exponentially decaying weight, i.e work in $H_a$ instead, we can again use this technique of looking at asymptotic operator, however I am not sure how to justify this or actually prove anything here.

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