Dear Biswanath,
an answer to your interesting question can be obtainied by referring to the differential equation encoding the quantum harmonic oscillator. It is well known that this equation is related to the Hermitian equation, hence to the Hermitian polynomials Hn(x), n=0,1,2,---- From this fact it results that the lowest energy of the quantum harmonic oscillator cannot be negative ! Since the quantum harmonic oscillator is ,in some sense, an elementary fundamental object in quantum mechanics, from this fact one has argued that quantum system cannot have negative quantum energies.
Unfortunately this is a completely wrong opinion !!!.
This has been explicitly proved in my quantum gravity theory, by considering the quantum super Yang-Mills PDEs. However, I would like to emphasize, that it does not necessitate to use my quantum gravity theory to be convicted that also quantum harmonic oscillator, considered inside the classical quantum mechanics (namely encoded by the Schroedinger equation) can have negative eigenvalues. Really the eigenvalue Schroedinger equation of the quantum harmonic oscillator, reduces to the Hermite equation, but this admits the Hermite polynomials as solutions when one assumes that eigenfunctions are polynomially bounded at infinity. On the other hand with different boundary conditions one can have also complex numbers as eigenvalues of suitable eigenfunctions. This is not a new result ! ... (Courant & Hilbert, Methods of Mathematical Physics, Volume I, Wiley-Interscience, 1953.)
Therefore, more than to ask whether there exists a transformation transforming positive eigenvalue in negative ones it is more interesting to consider that also the quantum harmonic oscillator can have complex eigenvalues and understand that this fact is related to suitable boundary value problems.
My congratulations for your intriguing question.
My best wishes,
Agostino
Dear agostino
Let me give a clear thought ,what is in my mind.simple harmonic oscillator under simultaneous non-hermitian transformation of coordinate and momentum with well behaved wave function with positive frequency of oscillation can have negative spectrum I.e -2n-1 ? Please give hints on your answer with little maths.
Biswanath
Dear Biswanath,
I have seen that you are interested to differential equations ... Therefore it should not be too difficult for you to relate the eigenvalue equation for the quantum harmonic oscillator to the Weber equation, and some hypergeometric functions. These are classical subjects ... A standard reference is the following:
Higher Transcendental Functions, I-II, Bateman Project Staff (A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi), Newy York, McGraw Hill, 1953-54.
Unfortunately, even if this subject refers to classical mathematics, it is impossible to resume it here in a form shorter than I just made.
All the best,
Agostino
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