Title:
Tutorial example - the screened one-dimensional "Coulomb" potential -1/sqrt(x^2+2)
The purpose of this exercise is
(1) become familiar with the popular "1d hydrogen" model
(2) study the effect of truncating the Coulomb-like tail
[TASK] Remove the turncation from the potential and observe its effect on
the bound state energies
[TASK] Try to determine a small basis to get the ground state accurate to 4 digits
Do not make the basis too large! What is the role of polynomial order?
# piece-wise specification of discretization with exponntial tails at both ends
Axis: name,nCoefficients,lower end, upper end,functions,order
X,20,-Infty,-20,polExp[0.5]
X,15,-20,-15,polynomial,15
X,60,-15, 15,polynomial,10
X,15, 15, 20,polynomial,15
X,20, 20, Infty,polExp[0.5]
# the function "trunc" serves to smoothly truncate a potential
# trunc[from,to]
# =1 for |Q| < from
# =0 for |Q| > to
# else 3rd order polynmial in Q with 0 derivatives at Q=from and Q=to
Operator: hamiltonian='0.5-'
# NOTE: it is essential that the the trunc-boundaries coincide with element boudaries
# although the code should work even if this is violated,
# the high order and with it efficienty is lost
# we will print the lowest eigenvalues and stop
Eigen: select=SmallReal[10]
Plot: axis,points,lowerBound,upperBound
X,201,-20.,20.