Letters in Mathematical Physics最新文献

We present two involutivity theorems in the context of Poisson quasi-Nijenhuis manifolds. The second one stems from recursion relations that generalize the so-called Lenard–Magri relations on a bi-Hamiltonian manifold. We apply these results to the closed (or periodic) Toda lattices of type (A_n^{(1)}), (C_n^{(1)}), (A_{2n}^{(2)}), and, for the ones of type (A^{(1)}_n), we show how this geometrical setting relates to their bi-Hamiltonian representation and to their recursion relations.

We introduce the quantum Berezinian for the quantum affine superalgebra (textrm{U}_q(widehat{mathfrak {gl}}_{M|N})) and show that the coefficients of the quantum Berezinian belong to the center of (textrm{U}_q(widehat{mathfrak {gl}}_{M|N})). We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of (textrm{U}_q(widehat{mathfrak {gl}}_{M|N})).

For a (*)-automorphism group G on a von Neumann algebra, we study the G-quasi-invariant states and their properties. The G-quasi-invariance or G-strongly quasi-invariance is weaker than the G-invariance and has wide applications. We develop several properties for G-strongly quasi-invariant states. Many of them are the extensions of the already developed theories for G-invariant states. Among others, we consider the relationship between the group G and modular automorphism group, invariant subalgebras, ergodicity, modular theory, and abelian subalgebras. We provide with some examples to support the results.

Both the Darboux transformation (DT) and Bäcklund transformation (BT) approaches of the modified Camassa-Holm (mCH) equation are restudied. The N-DT is constructed for the mCH equation in a simple and direct way. By extending the existing 1-BT and 2-BT, the N-BT of the mCH equation is obtained. It is argued that two multi-soliton solution formulae resulted from N-DT and N-BT are equivalent. Furthermore, the DT method is applied to calulate some explicit solutions which include a solution expressed in terms of trigonometric functions.

We consider discrete Schrödinger operators with real periodic potentials on periodic graphs. The spectra of the operators consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain periodic graphs of smaller dimensions called subcovering graphs. For example, rolling up a planar hexagonal lattice along different directions will lead to nanotubes with various chiralities. We describe connections between spectra of the Schrödinger operators on a periodic graph and its subcoverings. In particular, we provide a simple criterion for the subcovering graph to be isospectral to the original periodic graph. By isospectrality of periodic graphs we mean that the spectra of the Schrödinger operators on the graphs consist of the same number of bands and the corresponding bands coincide as sets. We also obtain asymptotics of the band edges of the Schrödinger operator on the subcovering graph as the "chiral" (roll up) vectors are long enough.

Any semigroup (mathcal {S}) of stochastic matrices induces a semigroup majorization relation (prec ^{mathcal {S}}) on the set (Delta _{n-1}) of probability n-vectors. Pick X, Y at random in (Delta _{n-1}): what is the probability that X and Y are comparable under (prec ^{mathcal {S}})? We review recent asymptotic ((nrightarrow infty )) results and conjectures in the case of majorization relation (when (mathcal {S}) is the set of doubly stochastic matrices), discuss natural generalisations, and prove a new asymptotic result in the case of majorization, and new exact finite-n formulae in the case of UT-majorization relation, i.e. when (mathcal {S}) is the set of upper-triangular stochastic matrices.

The Grover matrix of a graph G is a typical time evolution matrix of a discrete-time quantum walk on G. We treat the Grover matrix of a finite covering of G and present a decomposition formula for the determinant of it. Furthermore, we define an L-function of a graph with respect to the Grover matrix and present its determinant expression. As a corollary, we express the determinant of the Grover matrix of a covering of G as a product of its L-functions.

The (GL_{ell +1}(mathbb {R})) Hecke-Baxter operator was introduced as an element of the (O_{ell +1})-spherical Hecke algebra associated with the Gelfand pair (O_{ell +1}subset GL_{ell +1}(mathbb {R})). It was specified by the property to act on an (O_{ell +1})-fixed vector in a (GL_{ell +1}(mathbb {R}))-principal series representation via multiplication by the local Archimedean L-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group (GL_{ell +1}(mathbb {R})times GL_{ell +1}(mathbb {R})) by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group (Sp_{2ell +2}(mathbb {R})times Sp_{2ell +2}(mathbb {R})) by a Heisenberg Lie group.

We consider classical theories described by Hamiltonians H(p, q) that have a non-degenerate minimum at the point where generalized momenta p and generalized coordinates q vanish. We assume that the sum of squares of generalized momenta and generalized coordinates is an integral of motion. In this situation, in the neighborhood of the point (p=0, q=0), the quadratic part of a Hamiltonian plays a dominant role. We suppose that a classical observer can observe only physical quantities corresponding to quadratic Hamiltonians and show that in this case, he should conclude that the laws of quantum theory govern his world.

Let ((textbf{U}, textbf{U}^imath )) be a split affine quantum symmetric pair of type (textsf{B}_n^{(1)}, textsf{C}_n^{(1)}) or (textsf{D}_n^{(1)}). We prove factorization and coproduct formulae for the Drinfeld–Cartan operators (Theta _i(z)) in the Lu–Wang Drinfeld-type presentation, generalizing the type (textsf{A}_n^{(1)}) result from Przeździecki (arXiv:2311.13705). As an application, we show that a boundary analogue of the q-character map, defined via the spectra of these operators, is compatible with the usual q-character map. As an auxiliary result, we also produce explicit reduced expressions for the fundamental weights in the extended affine Weyl groups of classical types.