L^2 data analysis for resonances
Playground for different methods to find the complex energy of a resonance state (Eres = Er - i Gamma/2) from a series of calculations using square-integrable (L^2) wavefunctions.
I can think of four methods:
- complex scaling
- complex absorbing potentials
- stabilization method (Hazi-Taylor--not extrapolation to zero stabilization, that is poor man's ACCC)
- ACCC or RAC = complex extrapolation of the lowest root
All four separate again along finer distinctions, but let's not go there.
For all four, the Hamiltonian is parametrized, and in step 1 of the calculation, the low-energy spectrum is repeatedly computed for many values of the parameter. The ACCC/RAC method represents an exception, as it only requires the lowest state, but the tradeoff is that only "low-energy" resonances can be found. Step 1 then yields a data set: energy_j(parameter_value_k).
In step 2, the data set is analysed to yield Eres, and the four methods differ in both the way the Hamiltionian is parametrized in step 1 and in the way the resulting data set is analyzed in step 2.
This project aims at comparing the four different approches:
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The Jolanta model potential is used 1D: V(x) = (ax^2 - b) * exp(-cx^2) 3D: V(r) = (ar^2 - b) * exp(-cr^2) + 0.5l(l+1)/r**2
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A simple sine DVR of the Hamiltonian is used
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Each step goes into a Jupyter notebook