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I'm trying to solve this matrix to get a resulting function that depends on the variable Q (or if impossible, H3).

When I try to to that I get two results: if I try to solve for Q, it doesn't show anything and if I try to solve for H3 I get this message

This system cannot be solved with the methods available to Solve.

The problem is also that I'm not sure if it's even possible to solve this determinant with those recursive functions.

Here's the code:

m = 1;
h = 1;
a = 1;
b = 2;
n = 1;

V0 = -1;

V1 = 2;

V2 = 3;


k1[Q_] = Sqrt[2 m/h^2*(Q + V0)];

k2[Q_] = Sqrt[2 m/h^2*(Q - V1)];
H3[Q_] = Sqrt[2 m/h^2*(Q - V2)];

M = ({{SphericalBesselJ[n, 
      k1*a], -SphericalBesselJ[n, k2*a], -SphericalBesselY[n, k2*a], 
     0}, {k1*D[SphericalBesselJ[n, k1*r], r] /. 
      r -> a , -k2*D[SphericalBesselJ[n, k2*r], r] /. 
      r -> a, -k2*D[SphericalBesselY[n, k2*r], r] /. r -> a, 0}, {0, 
     SphericalBesselJ[n, k2*b], 
     SphericalBesselY[n, 
      k2*b], -SphericalHankelH1[n, I*H3*b]}, {0, -k2*
       D[SphericalBesselJ[n, k2*r], r] /. r -> b, 
     k2*D[SphericalBesselY[n, k2*r], r] /. 
      r -> b, -I*H3*D[SphericalHankelH1[n, I*H3*r], r] /. r -> b}});

MatrixForm[M];

Det[M];

Solve[Det[M] == 0, H3]
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  • $\begingroup$ What is H3? You appear to define it as a function but then later use it as a variable. You probably don' t want to do both. $\endgroup$ – bill s Dec 20 '18 at 23:01
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You can get an answer for H3 if you give values to k1 and k2. For instance:

Clear[H3]; m = 1; h = 1; a = 1; b = 2; n = 1; V0 = -1; V1 = 2; V2 = 3;
(* define M as above *)
Solve[Det[M /. {k1 -> 1, k2 -> 2}] == 0, H3]

gives H3 as a rather complicated root value.

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