Reviews in Mathematical Physics

We consider Bose–Einstein condensation (BEC) on graphs with transient adjacency matrix. We prove the equivalence of BEC and non-factoriality of the quasi-free state. Moreover, quasi-free states exhibiting BEC decompose into generalized coherent states. We review necessary and sufficient conditions that a quasi-free state is faithful, factor, and pure and quasi-free states are quasi-equivalent, including the papers of Araki and Shiraishi [1], Araki [2], and Araki and Yamagami [3]. Using their formats and results, we prove necessary and sufficient conditions that a generalized coherent state is faithful, factor, and pure and generalized coherent states are quasi-equivalent as well.

Utilizing the characteristic polynomial of Lax matrix for the Belov–Chaltikian (BC) lattice hierarchy associated with a $3\times 3$ discrete matrix spectral problem, we introduce a trigonal curve with three infinite points, from which we establish the associated Dubrovin-type equations. The essential properties of the Baker–Akhiezer function and the meromorphic function are discussed, that include their asymptotic behavior near three infinite points on the trigonal curve and the divisor of the meromorphic function. The Abel map is introduced to straighten out the continuous flow and the discrete flow in the Jacobian variety, from which the quasi-periodic solutions of the entire BC lattice hierarchy are obtained in terms of the Riemann theta function.

It turns out that a parametrization of degenerate density matrices requires a parametrization of $𝔉=U(n)/(U(k_1)\times U(k_2)\times \cdots \times U(k_m))$, $n=k_1+\cdots +k_m$ where $U(k)$ denotes the set of all unitary $k\times k$-matrices with complex entries. Unfortunately, the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods directly, but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra background and conclude with two concrete examples.

We develop a semiclassical approximation for the dynamics of quantum systems in finite-dimensional Hilbert spaces whose classical counterparts are defined on a toroidal phase space. In contrast to previous models of quantum maps, the time evolution is in continuous time and, hence, is generated by a Schrödinger equation. In the framework of Weyl quantization, we construct discrete, semiclassical Fourier integral operators approximating the unitary time evolution and use these to prove a Gutzwiller trace formula. We briefly discuss a semiclassical quantization condition for eigenvalues as well as some simple examples.