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Duality and Aholonomy 3
We again consider a particle moving on a line. We place the generalized point interaction at x = 0 as usual, but now demands the system to be time-reversal invariant. That condition translates into \xi = 0. We further require that the particle is bounced at the hard walls located at x = L1 and x = L 2 which are the assumptions made in order to discretize the entire spectra. The quantum mechanical "bounce back" is represented by Dirichlet condition \phi(L) = 0. Energy eigenvalue of the system E = k2 is obtained from |
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a sinkL1coskL2+a coskL1sinkL2+b/k sin kL1 sin kL2+d k cos kL1 coskL2
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（5.5）
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The system has three indenpendent parameters, and its full treatment isstill too tedius. We fix one of them, for example by setting c = c_0, and look at the energy as the function of remaining two parameters a and b
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E= E(a,b)
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（5.6）
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The energy surface thus obtained typically is an unremarkable object with random wavy wiggles, but at certain points, we find intriguing structures. One such example is depicted bellow. There is something singulat at the center around which all states form spiraling stair. Close examination reveals that theere are two disconnected stairs and the whole structure shows approximate 180 degree point symmetry. At the center of this double helix structure, we have b = 0 and a = 1/c_0, thus d becomes indefinite in (5.5) at which point there is no unique energy surface E(a,b).
At this stage, it would just be a wild speculation to contemplate on the possible usage of this structure. But it is still an entertaining thought to consider such mechanism as "quantum cycle", "quantum heat engine" etc. in the future world of nanoscale object control
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