Related papers: Isospectrality for quantum toric integrable system…
We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or…
The quantum dynamics of an electron in a uniform magnetic field is studied for geometries corresponding to integrable cases. We obtain the uniform asymptotic approximation of the WKB energies and wavefunctions for the semi-infinite plane…
We apply the approach developed in our previous papers to obtain examples of solutions to the inverse spectral problem (ISP) for the canonical Hamiltonian system. One of our goals is to illustrate connections of ISP with classical tools of…
This paper describes the reconstruction of the topological string partition function for certain local Calabi-Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations.…
We consider the question of what quantum spin chains naturally encode in their Hilbert space. It turns out that quantum spin chains are rather rich systems, naturally encoding solutions to various problems in combinatorics, group theory,…
We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non trivial way. These systems depend on a parameter t $\in$ [0, 1] and exhibit different behaviors…
We consider a model dissipative quantum-mechanical system realized by coupling a quantum oscillator to a semi-infinite classical string which serves as a means of energy transfer from the oscillator to the infinity and thus plays the role…
We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry…
We consider the quantum inverse scattering method for several mixed integrable models based on the rational SU(N) R-matrix with general toroidal boundary conditions. This includes systems whose Hilbert spaces are invariant by the discrete…
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global…
Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…
We describe the Poisson ideals and attached symplectic geometry of a cluster algebra with compatible Poisson structure. We apply these results to determine the spectrum of a quantum cluster algebra. As an application, we describe the…
We carry out an exact quantization of a PT symmetric (reversible) Li\'{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard…
Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no…
In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes",…
We discuss, within the simplified context provided by the polymeric harmonic oscillator, a construction leading to a separable Hilbert space that preserves some of the most important features of the spectrum of the Hamiltonian operator.…
We propose in this work a concept of integrability for quantum systems, which corresponds to the concept of noncommutative integrability for systems in classical mechanics. We determine a condition for quantum operators which can be a…
We review some recent results concerning integrable quantum field theories in 1+1 space-time dimensions which contain unstable particles in their spectrum. Recalling first the main features of analytic scattering theories associated to…
An integrable anharmonic oscillator is presumably simulable by a classical computer and therefore by a quantum computer. An integrable anharmonic oscillator whose Hamiltonian is of normal type and quartic in the canonical coordinates is not…
Essential properties of semiclassical approximation for quantum mechanics are viewed as axioms of an abstract semiclassical mechanics. Its symmetry properties are discussed. Semiclassical systems being invariant under Lie groups are…