let be a non-smooth potential , for example a linear combination of step functions

$$ \sum_{n=0}^{10}H(x-n) $$

my question is, for this potential would be Bohr-sommerfeld quantization formula valid ??

is there any resource where they apply bohr sommerfeld formula to non-smooth potentials

$$ 2\pi (n+\frac{1}{2})\hbar = \oint _{C}p.dq $$

  • $\begingroup$ Comment to the question(v1): OP's explicit example $V(x)=\sum_{n=0}^{10}H(x-n)$ does not have any bounded states (which would mean that the Bohr-Sommerfeld semiclassical quantization formula does not apply), unless one additionally imposes a lower bound for the $x$-coordinate. $\endgroup$ – Qmechanic Nov 19 '12 at 16:33
  • $\begingroup$ er wel for exqample a potential that tends to infinity , like $ [x] $ with Dirichlet conditions at $ x=0 $ , or for example any potential like $ \sum_{n=0}^{\infty}H(x^{2}-n^{2}) $ $\endgroup$ – Jose Javier Garcia Nov 19 '12 at 18:13

I believe it is valid without a continuity assumption, but (as always) only up to higher orders in $\hbar$.

  • $\begingroup$ you mean piecewise continous function ??, for example like an step function or similar ??, $\endgroup$ – Jose Javier Garcia Nov 19 '12 at 12:10
  • $\begingroup$ @JoseJavierGarcia: Yes. You can approximate it by a smooth function and then take a limit. $\endgroup$ – Arnold Neumaier Nov 19 '12 at 14:34

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