Abstract
We define a generalization of the Töplitz quantization, suitable for operators whose Töplitz symbols are singular. We then show that singular curve operators in Topological Quantum Fields Theory (TQFT) are precisely generalized Töplitz operators of this kind and we compute for some of them, and conjecture for the others, their main symbol, determined by the associated classical trace function.
In memory of Erik Balslev
from whom I learned so much
in mathematics and in physics
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Notes
- 1.
The construction works certainly also for conditions of the type, e.g..,
$$ (\tau (1-\tau ))^{\frac{\alpha |k|}{2}}\gamma _k(\tau )\in C^\infty ([0,1]), k=1,\dots ,N-1,\ 0\le \alpha \le 1 $$(or even more general ones), but since we don’t see any applications of these situation, we concentrate in this paper to the condition (1.5).
- 2.
by this we mean that \(\alpha \) is bounded on [0, 1] and \(C^\infty \) on any open subset of [0, 1].
- 3.
one can also say that \(\text{ Op}_{z}^{T}[{s}]={s}^{PS}\left( \text{ Op}^{APD}_{z}\left( \frac{\mathcal Z_z}{|z|}\right) , \text{ Op}^{APD}_{z}(\tau )\right) , \) where \({s}^{PS}\) is the pseudodifferential ordering of the trigonometric polynomial \({s}\), that is the one with all the \(\frac{\text{ Op}^{APD}_{z}(\mathcal Z_z)}{|z|}\) on the left.
- 4.
Note that in [12] we proven this type of result by another method since we wanted also to define a symbol inside the interior of \(\Sigma \) in the singular cases. Since we proved that in any coloring the three matrices are \(a\)-Töplitz operators, we don’t need such a direct result here.
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Paul, T. (2021). Symbolic Calculus for Singular Curve Operators. In: Albeverio, S., Balslev, A., Weder, R. (eds) Schrödinger Operators, Spectral Analysis and Number Theory. Springer Proceedings in Mathematics & Statistics, vol 348. Springer, Cham. https://doi.org/10.1007/978-3-030-68490-7_10
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