(8512) Gamow decay from a deep box via a delta barrier

Consider the decay of a particle of mass \(M\) in semi-1D geometry from a box of length \( a \). The potential is \( -V \) inside the box ( \( 0 < x < a \) ), and zero outside of the box ( \(x>a \) ). Additionally, there is a delta barrier \(u\delta(x-a)\) that slows down the decay. In the numerical part choose units such that \(M=V=1 \), and accordingly the dimensionless parameters of the problem are \(a\) and \(u\). The objective is to show the dependence of the spectrum on \(a\) and \(u\). Each eigenvalue \(E=E_n-i(\Gamma_n/2)\) is represented by a vertical bar at \(E_n\) of height \(\Gamma_n\).

(1) Write an expression for the logarithmic derivative \(q(E)\) of the interior solution at \(x=a+0\). Below it is more convenient to use \(k=\sqrt{2M(E-V)}\) instead of \(E\) as a free variable.

(2a) Explain why the equation \( q(E) = - \sqrt{-2ME} \) determines the bound states.

(2a) Explain why the equation \( q(E) = i \sqrt{2ME} \) determines the resonances.

(3) Find the eigenvalues numerically:

(3a) Display the results as a function of \(a\) for \(u=0\), and for an additional representative large value of \(u\).

(3b) Display the results versus \(u\) for a representative value of \(a\).

(4) Find analytical approximations:

(4a) Expand to linear order \(q(E) \approx C_n (k-k_n^{(0)})\), where \(k_n^{(0)}\) indicate the roots of the equation \(q(E)=0\).

(4b) Regarding the \(k_n^{(0)}\) as known, solve for the complex roots \(k_n \), and hence obtain expressions for \(E_n \) and \( \Gamma_n \).

(4c) To get explicit results, approximate \(k_n^{(0)}\) using either Dirichlet or Neumann boundary conditions.

(4d) Specify the conditions that justify each of the approximations in item 4c.

(5) Establish your approximations numerically (add analytical lines to the numerical plots). Please provide a Matlab or a Python file that produces the results.

Please note a related question Ex9280.