Suppose I have a non-self adjoint operator in a PDE which, being nonself adjoint, has deficiency indices in its spectrum and therefore requires an extension. Say that I solve this PDE numerically and want to analyze the graphical functions. Can these functions make some physical sense, although the inherent operator of the PDE is non-self adjoint?
I am really asking this because many rogue wave phenomena *appear* to be described by non self adjoint operators, and after transformation of their related PDE, physically sensible analytical and numerical solutions can always be gained. For instance, the Darboux transformation or the generation of a LAX pair is necessary to solve the NLSE, but is it because the NLSE is non-self adjoint and would otherwise give meaningless numerical results?
As an example:
Suppose I were to analyze the quartic Schrödinger equation:
(D2-V4)*Psi=E*Psi
where the quartic well potential replaces the quantum potential V2 from the regular Schrödinger equation. Here the operator (D2-V4) acting on the wavefunction Psi is non-self adjoint, and has deficiency indices in its spectrum (requires an extension).
However, if I analyze this equation as it is numerically, and try to generate its solutions for arbitrary values of E, I get plots as attached. These plots show decaying oscillatory behavior further and further away from the origin, which is similar to the standard Schrödinger under an electromagnetic field for particles. But can these results be used to describe a phenomenon described by the quartic equation or does one have to transform it first to a form that yields a self adjoint operator on Psi first?
Dear Dr. @Sergio Manzetti@
Greating!
The following links may be useful
http://rspa.royalsocietypublishing.org/content/royprsa/469/2154/20130019.full.pdf
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https://books.google.com.eg/books?id=9R9GAAAAQBAJ&pg=PA278&lpg=PA278&dq=numerical+solutions+to+a+PDE+with+a+non-self+adjoint+operator&source=bl&ots=Yz-XWu2c-b&sig=0M19NKkZbbzVtgonaJ-siuw7Arw&hl=ar&sa=X&ved=2ahUKEwijpPyuu-3dAhVNzRoKHVYPCpUQ6AEwCHoECAEQAQ
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https://www.sciencedirect.com/science/article/pii/019688589090007L
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https://books.google.com.eg/books?id=T5X6CAAAQBAJ&pg=PA58&lpg=PA58&dq=numerical+solutions+to+a+PDE+with+a+non-self+adjoint+operator&source=bl&ots=0HUGxTSKHt&sig=eZ4Nax53gLjgdL5VDzZmXKa9hD0&hl=ar&sa=X&ved=2ahUKEwijpPyuu-3dAhVNzRoKHVYPCpUQ6AEwA3oECAYQAQ
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with best wishes.
Dear DrThe basic legrange's identitty with a link of the algebraic topology..check that out?
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