Hearing the shape of a quantum boundary condition Giuliano Angelone https://arxiv.org/pdf/2204.10248.pdf A first negative answer has been given in 1992 by C. Gordon et al.,4 who constructed an isospectral pair of two-dimensional polygons, depicted in Fig. 1. In the case of two-dimensional domains with a smooth boundary, a positive result can be recovered by requiring some additional symmetries, see e.g. the work of S. Zelditch,5 but the general problem is still unsolved. " 2022 We remark that although a twisted torus is topologically indistinguishable from a non-twisted one, S1/Z2 being indeed homeomorphic to S1, a function defined on the torus is generally modified by the twist. The action of the twist, as well as its necessity, is highlighted in Fig. 3, where the twisted torus is compared with the non-twisted one and also with the “doubled” torus ̃Σ ∼= D × S1.
The geometry of the spectral
space Σ ..., whose representation is both continuous and one-to-one.
the corresponding Hamiltonians, which
are not symmetric under parity, are related by Eq. (43) and thus all “sound” the
same. Moreover, we found that the spectral space Σ has a non-trivial topology, being
a twisted solid torus.
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