Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

remove Papenbrock/1d integral #208

Merged
merged 1 commit into from
Aug 2, 2023
Merged

remove Papenbrock/1d integral #208

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
5 changes: 4 additions & 1 deletion src/Hamiltonians/HOCartesianEnergyConserved.jl
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -11,7 +11,10 @@ Indices `i,j,k,l` start at `0` for the groundstate.
This integral has a closed form in terms of the hypergeometric ``_{3}F_2`` function,
and is non-zero unless ``i+j+k+l`` is odd. See e.g.
[Titchmarsh (1948)](https://doi.org/10.1112/jlms/s1-23.1.15).
Used in [`HOCartesianEnergyConserved`](@ref).
This is a generalisation of the closed form in
[Papenbrock (2002)](https://doi.org/10.1103/PhysRevA.65.033606), which is is the special
case where ``i+j == k+l``, but is numerically unstable for large arguments.
Used in [`HOCartesianEnergyConserved`](@ref) and [`HOCartesianEnergyConservedPerDim`](@ref).
"""
function four_oscillator_integral_general(i, j, k, l; max_level = typemax(Int))
all(0 .≤ (i, j, k, l) .≤ max_level) || return 0.0
Expand Down
28 changes: 3 additions & 25 deletions src/Hamiltonians/HOCartesianEnergyConservedPerDim.jl
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -1,25 +1,3 @@
# TO-DO: optimise if large arguments are used
"""
four_oscillator_integral_1D(i, j, k, l; max_level = typemax(Int))

Integral of four one-dimensional harmonic oscillator functions,
```math
\\mathcal{I}(i,j,k,l) = \\int_{-\\infty}^\\infty dx \\,
\\phi_i(x) \\phi_j(x) \\phi_k(x) \\phi_l(x)
```
assuming ``i+j = k+l``. State indices `i,j,k,l` start at `0` for the groundstate.
Closed form taken from [Papenbrock (2002)](https://doi.org/10.1103/PhysRevA.65.033606).
Used in [`HOCartesianEnergyConservedPerDim`](@ref).
"""
function four_oscillator_integral_1D(i, j, k, l; max_level = typemax(Int))
!all(0 .≤ (i,j,k,l) .≤ max_level) && return 0.0
min_kl = min(k, l)
# SpecialFunctions.gamma(x) is faster than factorial(big(x))
p = sqrt(gamma(k + 1) * gamma(l + 1)) / sqrt(2 * gamma(i + 1) * gamma(j + 1)) / pi^2
s = sum(gamma(t + 1/2) * gamma(i - t + 1/2) * gamma(j - t + 1/2) / (gamma(t + 1) * gamma(k - t + 1) * gamma(l - t + 1)) for t in 0 : min_kl)

return p * s
end

"""
largest_two_point_box(i, j, dims::Tuple) -> ranges
Expand Down Expand Up @@ -103,8 +81,8 @@ first-order degenerate perturbation theory.
where the ``\\delta``-function indicates that the noninteracting energy is conserved along each
dimension.
The integral ``\\mathcal{I}(a,b,c,d)`` is of four one dimensional harmonic oscillator
basis functions with the restriction that energy is conserved in each dimension,
see [`four_oscillator_integral_1D`](@ref).
basis functions, see [`four_oscillator_integral_1D`](@ref), with the additional restriction
that energy is conserved in each dimension.

# Arguments

Expand Down Expand Up @@ -169,7 +147,7 @@ function HOCartesianEnergyConservedPerDim(
bigrange = 0:M-1
# case n = M is the same as n = 0
vmat = reshape(
[four_oscillator_integral_1D(i-n, j+n, j, i; max_level = M-1)
[four_oscillator_integral_general(i-n, j+n, j, i; max_level = M-1)
for i in bigrange, j in bigrange, n in bigrange],
M,M,M)

Expand Down
Loading