Can I Solve This Complicated Equation with a Dirac Delta Function?

[Goddar]

Hi there, my version of Mathematica may be too old and I'm not finding this one by hand so any help would be appreciated:
ψ''(z)=[k²/4 –M² –kδ(z)]ψ(z),
where δ(z) is the Dirac delta, k and M constants.

i can solve the same equation without the M^2 term by exp(k|z|/2), but this one proves to be much more complicated.

Please note that this is part of a problem where solving the D.E. is actually not the issue: I'm supposed to use a software or anything that helps so I'm really just trying to get the answer.. thanks!

 

[Bill Simpson]

Version 9.0.1 DSolve can't crack it.

Perhaps this will help

In[1]:= k = 1; m = 2;
f = s[z] /. NDSolve[{s''[z] == (k^2/4 - m^2 - k DiracDelta[z]) s[z], s[0] == 0, s'[0] == 1/2}, s[z], {z, 0, 10}]

Out[2]= InterpolatingFunction[{{0., 10.}}, <>][z]}

In[3]:= Plot[f, {z, 0, 10}]

Out[3]= ...PlotSnipped...

 

[Goddar]

Thank you Bill, I'm going to look into this "interpolating function"...
Actually, exp[–k|z|/2]cos(Mz) almost works but i pick up an extra term in the D.E.; maybe an infinite series would do the job, i just can't find it so far...

 

[Thaakisfox]

you don't need Mathematica or anything, just solve the simple second order DE for z<0 and z>0 (the delta function term is zero here) and then match them at z=0. The function itself should be continuous and then there is a jump in the derivative at z=0 due to the delta function.

 

[Goddar]

Thank you, you're right. It's good to remember what these functions really are sometimes!..
I'm going to try that.