Revisiting double Dirac delta potential
  We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ($T=1$). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, $R(E)$] at energy $E=0$ is non-intuitive. However, earlier this has been found and called "threshold anomaly". Here we show that it is a critical phenomena and we can have $ 0 \le R(0)<1$ when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate ($\psi(x)=Ax+B$) of delta well between two symmetric rigid walls for $R(0)=0$. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently. 
Zafar Ahmed, Sachin Kumar, Mayank Sharma, Vibhu Sharma
quant-ph