It is claimed that Arthur C. Lunn at the University of Chicago found the Schrödinger equation of the hydrogen atom in 1921, but the paper was rejected by the Physical Review. At the beginning of the unpublished and then lost paper, Lunn used the Planck relation $E = h \nu $ to justify the de Broglie relation $\lambda=h/p$ and the wave nature of matter. Following that, Lunn obtained the hydrogen spectrum by solving the Schrödinger equation as an eigenvalue problem.
This information can be found in Lunn's biographical notes:
"The beginning of the Quantum mechanics paper, which Sam remembers, started with the equation $E=\hbar \omega$ and pointed out that the relativistic 4-vector must be completed to make $p=\hbar k$, i. e., the de Broglie result. The last part of the paper, which Sam did not read but which he heard about and which Weinberg remembered, went on to obtain the H energy spectrum by solving an eigenvalue problem, i.e., the Schrödinger result. The paper was rejected by Physical Review, with the referee (Fulcher) insisting that it was meaningless abstraction. Lunn then resigned as Associate editor of PR, and was replaced with Fulcher. This was either at the end of 1921 or 1922."
That is, Lunn's revolutionary idea was rejected. On the other hand, Arthur Compton's scattering theory, that described the particle nature of light in accordance with Einstein’s light quantum hypothesis, was published in 1923 by the Physical Review. It seems really strange that Lunn's work was not reconsidered, especially when it provided the correct result. Of course, Lunn, with his troubled mind, was part of the problem. People were also wholly ignorant that electron scattering experiments can be explained by the de Broglie relation. One such article on electron scattering by Davisson and Kunsman appeared in the Physical Review in 1922.
In any case, one source tells that "he had treated particles as expressions of 'beat' phenomena - that is, discrete solutions of wave equations". That might describe the usual quantum mechanical wave packets, but how did Lunn derive the Schrödinger equation? Could it be that he simply modeled the hydrogen atom as a vibrating system that obeys the well-known Helmholtz (wave) equation as an eigenvalue problem? It can be argued that this is the most obvious approach to the problem, especially if the topic of Lunn's paper is the matter wave hypothesis and its immediate applications.
We may assume that in his paper Lunn considered the hydrogen spectrum on the basis of the Bohr atomic model that describes the energies of stationary states and the frequencies of the radiation emitted in transitions between them. The model is successful in obtaining the negative energy levels $E$ that have been confirmed in empirical studies. If the wave nature of electrons is taken into account, the stationary states are replaced by stationary waves. That is, in 1924 de Broglie hypothesized that a stable electron orbit $nh=2\pi rp$ in Bohr's model is possible only if the electron wave ($p=h/\lambda$) fits continuously on the periodic perimeter of a circle, i.e., $n\lambda = 2\pi r$. This is the de Broglie-Bohr model where electrons orbit the nucleus as stationary waves, each orbit associated with a different energy level (angular) frequency $\omega =2\pi \nu = 2\pi E/h$ and wavelength $\lambda$. Needless to say, a more realistic model of the hydrogen atom should be three-dimensional and allow the hypothesized stationary waves to vibrate more freely.
Therefore, motivated by the matter wave hypothesis, the obvious task is to find a suitable wave equation to describe the electron, that is bound to the nucleus of hydrogen atom, and thus the collection of stationary wave solutions that are associated with the energy level frequencies $\omega = 2\pi E/h$, as correctly predicted by the old Bohr model. It is well-known that for vibrating systems, the Helmholtz equation allows to find all the natural frequencies, each one associated with a different stationary wave pattern that can store vibrational energy. In the atomic case, these stationary wave solutions must be smooth and vanish at the infinity, cf. vibrations of an infinitely large circular membrane. A vibrating atomic system may not be visualisable but the physical motivation is clear.
Consequently, we may assume that Lunn used the good old Helmholtz equation $\nabla^2 f = -k^2 f$, where $k=\omega /v_p = 2\pi \nu/v_p=2\pi /\lambda$. As we already know, this is an eigenvalue problem where the frequencies correspond to eigenvalues that then correspond to eigenfunctions. Via the de Broglie relation, we have $k^2 = 8 \pi^2m/h^2\cdot(p^2/2m) =8 \pi^2m/h^2\cdot(E-V)$. We note that for every solution, the frequency $\omega$ is constant but phase velocity $v_p$ and wavelength $\lambda$ are position dependent due to $V(x)$. Finally, by adding the Coulomb potential, we obtain the celebrated time-independent Schrödinger equation of the hydrogen atom:
$$\nabla^2 f =- \frac{8 \pi^2m}{h^2}(E+ \frac{e^2}{4\pi \epsilon_0 r})f.$$
