PT-invariant periodic potentials with a finite number of band gaps
- Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005 (India)
We obtain the band edge eigenstates and the midband states for the complex, generalized associated Lame potentials V{sup PT} (x)=-a(a+1)m sn{sup 2} (y,m)-b (b+1)m sn{sup 2} (y+K(m),m)-f (f+1)m sn{sup 2} (y+K(m)+iK{sup '} (m),m)-g(g+1)m sn{sup 2} (y+iK{sup '}x(m),m), where y{identical_to}ix+{beta}, and there are four parameters a, b, f, g. By construction, this potential is PT-invariant since it is unchanged by the combined parity (P) and time reversal (T) transformations. This work is a substantial generalization of previous work with the associated Lame potentials V(x)=a(a+1)m sn{sup 2}(x,m)+b(b+1)m sn{sup 2}(x+K(m),m) and their corresponding PT-invariant counterparts V{sup PT}(x)=-V(ix+{beta}), both of which involving just two parameters a,b. We show that for many integer values of a,b,f,g, the PT-invariant potentials V{sup PT}(x) are periodic problems with a finite number of band gaps. Further, using supersymmetry, we construct several additional, complex, PT-invariant, periodic potentials with a finite number of band gaps. We also point out the intimate connection between the above generalized associated Lame potential problem and Heun's differential equation.
- OSTI ID:
- 20699337
- Journal Information:
- Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 8 Vol. 46; ISSN JMAPAQ; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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