programming - Eigen energies of 2D time independent Schrödinger equation

Firstly, I am aware that a similar question of this type was asked, but it was not helpful as it was only one dimensional (so please don't mark my question as a duplicate)

The trouble I am having is using ParametricNDSolveValue to solve my Schrödinger PDE. Here is what I have:

ℏ = 1;
m = 1;
Δ = 10^-3;

k = 1;
X = 70;
V[x_, y_] := -k/Sqrt[1 + x^2 + y^2]
Sol = ParametricNDSolveValue[{-ℏ^2/(2 m) (D[ψ[x, y], {x, 2}] + D[ψ[x, y], 
      {y, 2}]) + V[x, y] ψ[x, y] == Etr ψ[x, y], ψ[x, -X] == Δ, ψ[x, X] == Δ, 
      ψ[-X, y] == Δ, ψ[X, y] == Δ}, ψ[0, 0], {x, -X, X}, {y, -X, X}, {Etr}]

It seems to work okay initially, but when I try to find roots of this function ie:

FindRoot[Sol, {-0.45}]

I get values that are inconsistent with my solution in polar/cylindrical coordinates. That solution gave me proper values and good solutions (because it was done in essentially 1 dimension) If you do the following plot:

EList = Range[-0.46, 0, 0.0005];
ETable = ParallelTable[{EList[[i]], Sol[EList[[i]]]}, {i, 1, 
    Length[EList]}];
ListPlot[ETable]

The result is terrible. Does anyone have any idea as to how to solve this issue, or whether it is a bug or not?

Comments

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I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...

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calculus and analysis - Integral of the Sinc product

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