Implicit region is not "valid" in ParametricNDSolveValue function?

Question

I am trying to solve a geometric problem with relation to my Schrödinger equation and its boundaries. Here is my code:

Clear["Global`*"];
m = 1;
ℏ = 1;
k = 1;
V = -k/Sqrt[1 + x^2 + y^2];

A = 8;
Δ = 10^-3;
SE[Etr_] := -ℏ^2/(2 m) \!\(
\*SubsuperscriptBox[\(∇\), \({x, y}\), \(2\)]\(ψ[x, y]\)\) +
V ψ[x, y] - Etr ψ[x, y] == 0

Ω = ImplicitRegion[x^2 + y^2 <= A^2, {x, y}];
BC = DirichletCondition[ψ[x, y] == Δ, 
   x^2 + y^2 == A^2];

Clear[Sol]
Sol = ParametricNDSolveValue[{SE[Etr], BC}, ψ[0, 
   0], {x, y} ∈ Ω, {Etr}]

Then when I insert a trial value for Etr, like so:

Sol[-0.5]

it gives a value but returns the following error:

ParametricNDSolve::femnr: "{x$54658,y$54659}\ [Element]ImplicitRegion[x$54658^2+y$54659^2<=64,{x$54658,y$54659}] is not a valid region specification."

Does anyone know how to fix this?

Answer 1

Try this:

    Clear["Global`*"];
m = 1;
\[HBar] = 1;
k = 1;
V = -k/Sqrt[1 + x^2 + y^2];

A = 8;
\[CapitalDelta] = 10^-3;
SE[Etr_] := -\[HBar]^2/(2 m) \!\(
\*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(\[Psi][x, y]\)\) +
   V \[Psi][x, y] - Etr \[Psi][x, y] == 0

\[CapitalOmega] = Disk[{0, 0}, A];
BC = DirichletCondition[\[Psi][x, y] == \[CapitalDelta], 
      x^2 + y^2 == A^2];

Clear[sol]
sol = ParametricNDSolve[{SE[Etr], 
   BC}, \[Psi], {x, y} \[Element] \[CapitalOmega], Etr]

I only replaced the definition of the domain Omega from the ImplicitRegion to simply a Disk, since Mma complained. All the rest is the same. After that you can visualize the solution as:

 Manipulate[
 Plot3D[Evaluate[\[Psi][Etr][x, y] /. sol], {x, 
    y} \[Element] \[CapitalOmega] , PlotRange -> All],
 {Etr, -0.6, 0}]

which looks like the image below

I do not know though, if this gives somehow the solution of the Scrödinger equation. It would be interesting to know, how do you cope with it?

Have fun!