Given the Hamiltonian for the the harmonic oscillator (HO) as
$$
\hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,,
$$
the Schroedinger equation can be reduced to:
$$
\left[
\frac{d^2}{dz^2}-\left(\frac{z^2}{4}+a\right)\right]\Psi=0~,
$$
where $a=-\frac{E}{\hbar\omega}$, $z=\sqrt{\frac{2m\omega}{\hbar}}$.
Now, the two independent solutions to this equation are the Wittaker's functions (Abramowitz section 19.3., or Gradshteyn at the beginning, where he defines the Wittaker's functions) $D_{-a-1/2}(z)$ and $D_{-a-1/2}(-z)$.
Apparently, there is no constraint on the values for $a$. In Abramowitz, especially, there is written "both variable $z$ and $a$ can take on general complex values".
Therefore my first question is:
Let us fix $a=i$ and let us therefore take the Wittaker's function $D_{-i-1/2}(z)$. This functions is solution of the time independent Schroedinger equation, and, therefore, is an eigenfunction of the ho hamiltonian. Since its value for the parameter $a$ is $i$, it follows that its eigenvalue $E$ must be $E=-i\hbar\omega$. However, this result is contradictory, since the hamiltonian must have only real eigenvalues, since it is hermitian. What do I do wrong?
My second question is:
Since the functions $D_n(z)$ form a complete set for $n$ positive integer with zero, I can expand my function $D_{-i-1/2}(z)$ onto the basis set $D_n(z)$.
$$
D_{-i-1/2}(z)=\sum_n C_n D_n(z)~.
$$
But, evidently, if $D_{-i-1/2}(z)$ is itself an eigenfunction with a different eigenvalue with respect to any of the $D_n(z)$, the expansion above does not make sense.
This question is somewhat correlated to the previous one. So, I believe I do something
wrong which is in common to both of them.
Best Answer
"the hamiltonian must have only real eigenvalues, since it is hermitian." - only for physically meaningful systems. You have put $a=i$, meaning $m$ or $\omega$ is negative - or maybe in some bizarre alternate universe, $\hbar$. Or $m$ and $\omega$ could each be imaginary. Boundary conditions must be met. You are, in effect, exploring solutions that don't persist in time with a $exp(i\omega t)$ dependence, but decaying (or growing) exponentially. A quantum system's wavefunction shouldn't be allowed to do that!