Density of states with changing potential floor I wish to calculate the density of states for a 1D finite potential well, for the scattering states, meaning $E>0$.
I have a potential of the form:
$$
V(x)=\begin{cases}
-V_0 & |x|<a\\
0 & else
\end{cases}
$$
Obviously my spectrum is continuous (free waves), so I would expect something similar to:
$$
g(E)={1\over2\pi}\sqrt{2m\over E-E_{min}}
$$
due to the fact the my system is 1D.
My problem is that the $E_{min}$ term has spatial dependence due to the well. 
Does it make sense to have spatial dependence in the DOS?
If so, is the DOS a piecewise function? 
If not, what am I doing wrong?
By the way my motivation for this, is to calculate the probability that the electron will escape from the well due to perturbation, using Fermi Golden Rule.
 A: I figured it out, and decided to post it here so someone else might benefit from it if he comes across this issue.
It seems that I was looking at the question from the wrong point of view. Instead of taking the known expression which I posted in the question, one should analyze the system from scratch to get the answer.
Basically, we know that the number of states is the sum of all accessible levels:
$$
N(E)=\sum_n\theta(E-\epsilon_n)
$$
In our case we are interested in scattering states ($E>0$) with a continuous spectrum. However the negative energies which are also allowed due to to the changing potential floor are not part of this continuous spectrum.
In fact as we know the spectrum within the well is discrete.
Therefore the correct way to evaluate the number of states of the system, would be to sum up the discrete and continuous spectra in the following manner:
$$
N(E)=\sum_{n\in \text{ bound states}}^N\theta(E-\epsilon_n)+\int_0^\infty \underbrace{\frac{dN}{dE}}_{g_f(E)}dE
$$
In the last expression I assumed we have $N$ bound states in the well. $g_f(E)$ denotes the free wave density of states in 1D.
From this we can derive the density of states of the system to be:
$$
g(E)=\sum_{n\in \text{ bound states}}^N\delta(E-\epsilon_n)+g_f(E)=\sum_{n\in \text{ bound states}}^N\delta(E-\epsilon_n)+\theta(E)\frac{1}{\hbar}\sqrt{\frac{2m}{E}}
$$
