I am talking about the most famous paper of Heisenberg, which I know from the translation of van der Waerden (Sources in Quantum mechanics, North Holland, 1967). After introducing matrix mechanics Heisenberg writes (p. 12):
one could regard our equations as satisfactory...if it were possible to show that this solution agrees with quantum mechanics relationships which are known at present
His main examples are the anharmonic oscillators. On pp. 272-273 he considers a quartic oscillator whose classical equation is $$x''+\omega_0^2x+\lambda x^3=0,$$ which corresponds to the potential $\omega_0^2x^2/2+\lambda x^4/4$ in the Schrodinger equation. He derives two terms of the perturbative series of energy in powers of the Planck constant. Then he says:
This energy can be also determined using the Kramers-Born approach... The fact that one obtains exactly the same result seems to me to furnish remarkable support for the quantum-mechanical equations which have been taken here as a basis
My question: What exactly does he mean by Kramers-Born approach to anharmonic oscillator, what is the reference? (There are some references on Kramers and Born in the beginning of the paper, I looked at them, they do not seem to be relevant. Neither van-der-Waerden's commentary is helpful. Neither the book of Mehra and Rechenberg, Historical development of quantum theory helps.
So I repeat my question: What does Heisenberg compare his result with? This comparison makes him believe that his quantum mechanics is correct. This seems to be the crucial part of the paper.