is Bohr-sommerfeld formula valid if the potential is non-smooth?

Question

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

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

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. — Qmechanic, Nov 19 '12 at 16:33
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}) $ — Jose Javier Garcia, Nov 19 '12 at 18:13

score 2 · Answer 1 · answered Nov 19 '12 at 12:00

2

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

answered Nov 19 '12 at 12:00

Arnold Neumaier

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

1 Answers1