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I have this question,when i was told QM they taught me some axioms:(1D)

The wave function of the particle is given by the differential equation:

[tex]i\hbar\frac{d\psi}{dt}=\frac{-\hbar^{2}}{2m}D^{2}\psi+V(x)\psi [/tex] with D=d/dx

and that the eigenfunctions of [tex]H\phi=E_{n}\phi[/tex] are all orthogonal and are on L^2(R) function space....

i have discussed in other forum a method to obtain RH by assuming that a Hamiltonian have its energies being the eigenvalues of a certain function f(x) my question in this case is if:

a)does the Schroedinguer equation have always a solution independent of what the potential V is?.,let,s suppose that potential is dicontinous everywhere (impossible but mathematically true).

b)are always the eigenfnction of the Hamiltonian on the space L^2(R)

c)can we always say that exist a potential V so the energies of the system are the roots of a certain function f(x)?...

i have some discussion with mathematicians there saying that ...

The wave function of the particle is given by the differential equation:

[tex]i\hbar\frac{d\psi}{dt}=\frac{-\hbar^{2}}{2m}D^{2}\psi+V(x)\psi [/tex] with D=d/dx

and that the eigenfunctions of [tex]H\phi=E_{n}\phi[/tex] are all orthogonal and are on L^2(R) function space....

i have discussed in other forum a method to obtain RH by assuming that a Hamiltonian have its energies being the eigenvalues of a certain function f(x) my question in this case is if:

a)does the Schroedinguer equation have always a solution independent of what the potential V is?.,let,s suppose that potential is dicontinous everywhere (impossible but mathematically true).

b)are always the eigenfnction of the Hamiltonian on the space L^2(R)

c)can we always say that exist a potential V so the energies of the system are the roots of a certain function f(x)?...

i have some discussion with mathematicians there saying that ...

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