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Let <i>A</i> and <i>B</i> be non-negative self-adjoint operators in a separable Hilbert space such that their form sum <i>C</i> is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the <i>L</i> ²-norm, that is, one has <div class="gsh_dspfr"> \lim_{no+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} ight)^nh - e^{-itC}hight\|^2dt = 0

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Using the complex coloring method, we present the graphs of the quantum dilogarithm function Gb(z) and visualize its analytic and asymptotic behaviours. In particular we demonstrate the limiting process when the modified Gb(z)→Γ(z) as b→0. We also survey the relations of Gb(z) with different variants of the quantum dilogarithm function.

We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 smooth loops of a fixed length L, formally given by (x) with &gt;0. It is shown that the ground state of this operator is locally maximized by a circular . We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of .