# eigenvalue probelm with an implicit defined function

let be the eigenvalue problem

$$-y''(x)+q(x)y(x)=z_{n}y(x)$$

with boundary conditions $y(0)=0=y(\infty)$

my problem is that i do not know exactly what $q(x)$ is i know it only implicitly i mean i know its inverse $q^{-1}(x)= g(x)$

however $g(x)$ is hard to invert or it has no invesrse or has several branches so my idea is if we can define a new variable

$$x= q^{-1}(u)=g(u)$$

if we define a new fnction $F(u)=y(g(u))$ the new ODE turns into

$$-F''(u)g'(u)-g''(u)F'(u)+uF(u)=z_{n}F(u)$$ (1)

so know is hould numerically solve this equation which involvers $g(u)$ the inverse of the potential but not $q(x)$ is my reasoning right ??

this may work for example with the potential defined implicitly by

$$q^{-1}(x)= x+ sin(x)+cos(sin(x))$$

because the inverse $q(x)$ can be only estimated numerically

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