Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the indicator function. A particle of energy $E$ is incident from the left, giving eigenfunctions of the form $e^{\pm ik x},$ where $k =\sqrt{2mE}/\hbar,$ or $e^{\pm Kx}$. Carefulling picec these together at the boundary of the barrier gives a family of eigenfunctions or the Schrodinger operator $$ H=- \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2 }+ V(x). $$
Solutions of the Schrodinger equation can be obtained by decomposing w.r.t. these functions. The mechanism is similar to Fourier transform.
However, there is one caveat: the eigenfunctions are not unitary characters of the additive group on $\mathbb R.$ So Fourier transform on locally compact Abelian group does not work.
I know that this is where spectral theorem applies - however the spectral theorem is not so precise - it just gives me the existence of unitary equivalence to a multiplication operator; it does not tell me the exact form of this equivalence.
How can we prove rigorously that any solution of $H \psi = i\hbar \partial_t \psi$ is a "linear combination" of eigenfunctions above, in the form of an integral? i.e. if $\psi_E$ is the eigenfunction corresponding to eigenvalue $E,$ then do we have some relations like $$ \hat u(E) = \int \psi_E(x)^* u(x) dx, u(x) = \int \psi_E(-x)^* \hat u(E) dE? $$
