Quantization is an [art, not a functor](https://physics.stackexchange.com/questions/701695/what-does-it-mean-to-quantise-a-system), as comments reminded you. Here is a path to start thinking about your problem. You may arbitrarily *choose* the quantization $$ \hat H=c \hat p {1\over \hat x} \hat p, $$ which has your classical Hamiltonian as its classical limit, and is manifestly ***Hermitean***, among 37897 similar hamiltonians with this property. (The other answer chooses $H={c\over 2}\left ( \hat p ^2 {1\over \hat x} + {1\over \hat x} \hat p ^2 \right )$, instead, complicating the footnote ODE.) Your proposal is *not* Hermitean. Working in the coordinate representation, equivalent to the momentum one by Fourier transformation (the representation has little to do with quantization!), you have $$ -c\hbar^2 \left({1\over x} \psi’\right )’=E\psi, $$ which you might solve$^\natural$, and Fourier transform, as you have learned in elementary QM; any system may be Fourier transformed, including those with a brick-wall step function potential. You may impose the restriction of variables by a boundary condition, just as you handle the solutions of the infinite square well free hamiltonian. ----------------------------------------- $^\natural$<sub> Try $$ x y''-y'+b x^2 y =0 $$ in Wolfram α. </sub>