# Solving an equation involving a determinant (including spherical recursive functions)does not compute

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]

• 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. – bill s Dec 20 '18 at 23:01

Clear[H3]; m = 1; h = 1; a = 1; b = 2; n = 1; V0 = -1; V1 = 2; V2 = 3;