Do Delta Potentials Allow for Even Solutions in Quantum Mechanics?

[G01]

Homework Statement

Consider the potential:

$$V(x) = \alpha\delta(x)$$ -a<x<a
$$V(x) = \infinity$$ |x|>a


Analyze the even and odd solutions separately, and find the allowed energies.

Homework Equations

The Attempt at a Solution

So far, I looked at the even solutions:

$$\psi(x)=A\cos(kx)$$ 0<x<a
$$\psi(-x)$$ -a<x<0

With this solution, Acoskx must equal zero at the delta barrier correct?
Since the only way this could happen is for A=0, am I to assume the even solutions don't exist?

Am I going about this correctly? Thanks for any help you can offer.

 

[G01]

I think I may be confused on what they mean by "the even and odd solutions."

 

[Dr Transport]

Solutions which are odd and even under coordinate inversion...

 

[G01]

Ok I think I get that much, did I at least start this problem the correct way? Like, do I have the correct form of the even solutions?