Related papers: Intertwining operator for $AG_2$ Calogero-Moser-Su…

We discuss two polynomial bi-Hamiltonian structures for the generalized integrable Chaplygin system on the sphere S^2 with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation, the…

It has long been known that two-point functions of conformal field theory (CFT) are nothing but the integral kernels of intertwining operators for two equivalent representations of conformal algebra. Such intertwining operators are known to…

We review the higher-order supersymmetric quantum mechanics (H-SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first-order SUSY transformations are used to derive in a simple way the…

We study the algebra of invariant differential operators on a certain homogeneous vector bundle over a Riemannian symmetric space of type $A_2$. We computed radial parts of its generators explicitly to obtain matrix-valued commuting…

We construct and normalise intertwining operators at the level of Hilbert modules describing the principal series of SL(2). Normalisation is achieved through the use of a Fourier transform defined on some homogenous space and twisted by a…

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We…

We consider scalar two-dimensional quantum field theories with the factorizing S-matrix which has poles in the physical strip. In our previous work, we introduced the bound state operators and constructed candidate operators for observables…

We generalize the standard first-order intertwining relationship of supersymmetric quantum mechanics in order to include simultaneous scaling transformations in both the original Hamiltonian and the intertwining operator. It is argued that…

We review the approach of generalized permutator to produce a class of integrable quantum Hamiltonians, as well as the technique of Sutherland species (SS) to map a subclass of it into solvable spinless fermions models. In particular, we…

We consider a class of hyperplane arrangements $\mathcal A$ in ${\mathbb C}^n$ that generalise the locus configurations of \cite{CFV}. To such an arrangement we associate a second order partial differential operator of Calogero-Moser type,…

For two self-adjoint operators $H,A$ we show that a general commutation relation of type $[H,\mathrm{i}A]=Q(H)+K$, in addition to regularity of $H$ and Kato-smoothness of $K$, guarantee pointwise in time decay rates of diverse order. The…

The key concept discussed in these lectures is the relation between the Hamiltonians of a quantum integrable system and the Casimir elements in the underlying hidden symmetry algebra. (In typical applications the latter is either the…

Let $\Phi:V\to V\otimes U$ be an intertwining operator between representations of a simple Lie algebra (quantum group, affine Lie algebra). We define its generalized character to be the following function on the Cartan subalgebra with…

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can…

Using the Baker-Akhiezer function technique we construct a separation of variables for the classical trigonometric 3-particle Ruijsenaars model (relativistic generalization of Calogero-Moser-Sutherland model). In the quantum case, an…

Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex…

We investigate the quantum properties of 1D quantum systems whose classical counterpart presents intermittency. The spectral correlations are expressed in terms of the eigenvalues of an anomalous diffusion operator by using recent…

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the…

We construct commuting transfer matrices for models describing the interaction between a single quantum spin and a single bosonic mode using the quantum inverse scattering framework. The transfer matrices are obtained from certain…

We consider quantum Hamiltonians of the form $H = H_0 - U \sum_j \cos(C_j)$ where $H_0$ is a quadratic function of position and momentum variables $\{x_1, p_1, x_2, p_2,...\}$ and the $C_j$'s are linear in these variables. We allow $H_0$…