# How can we enhance the performance of NDEigensystem?

I am trying to reproduce eigenvalues in Tabl2 II in this paper (also available on the arxiv) for B=-12 but the results are slightly different from those in the paper. How can we enhance the performance of the NDEigensystem. The main objective is to solve Schrodinger Equation with a Deformed Morse-like potential

$$$$[-\frac{1}{2} \frac{d}{dx^2}+V(x)]\psi(x)=E\psi(x)$$$$ where $$$$V(x)=A(e^{\lambda x}+q)^{-2}+B(e^{\lambda x}+q)^{-1}-\frac{A+qB}{q^2}$$$$.

Here is my code to solve this

systm[B_] := Module[{A = 2.0, c = 1.0, q = 0.2, λ = 1.0},
{vals, funs} =
NDEigensystem[{-1/
2 ψ''[
x] + (A (Exp[λ x] + q)^-2 +
B (Exp[λ x] + q)^-1 - (A + q B)/q^2) ψ[x],
DirichletCondition[{ψ[x] == 0}, True]}, {ψ[
x]}, {x, -10., 5.}, 5,
Method -> {"SpatialDiscretization" -> {"FiniteElement", \
{"MeshOptions" -> {MaxCellMeasure -> 0.01}}}}];
Return[vals]]



the first 5 eigenvalues are

 systm[-12]

{0.0870463, -0.443629, 0.519853, 1.1084, -1.48047}


if we check only 2nd and 5th are correct which represent ground and first excited states

Plot[Evaluate[ funs[[#]]], {x, -10, 5}, PlotRange -> All,
PlotStyle -> Red] & /@ Range[5]


These are the ground and first excited states.

compared to the paper these eigenvalues are {-0.354453319,-1.348987620 } Note that in the paper they are positive because they add a minus sign as mentioned in the caption. They also used different methods to evaluate that as in this table

• Some comments: you are talking about the second and sixth eigenvalues but you only show five solutions. I am expecting an honest typo, right? From checking the paper-unless I am completely blind or missing something obvious-none of the values agree. In the paper all the values are positive. Also, it seems that in the 1-dimensional Schrodinger equation you set $E=0$ but I did not find this detail from a quick read of the paper. Finally, it would be very beneficial to explain what you are trying to do rather than providing only the code. Code is good. Code plus formulae better :)
– bmf
Jan 15 at 16:33
• @bmf they mentioned that they add the minus sign for eigenvalues, see table caption in the first line. Yes, I calculated only 5 but 2 are correct which are the first two ground states in the table. we can go further and check others but they are all slightly larger compared with MMA. $E\neq0$, I just include the operator Jan 15 at 16:40
• thanks for extra comments. I am probably already too tired to read another paper for today, sorry. but perhaps, you could add some more math details to your post to make it self-consistent which was my main point behind my questions :)
– bmf
Jan 15 at 16:42