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We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to…
This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I (also uploaded in arXiv.org) lead to a new type of the Abel map…
In this work we study the general system of geodesic equations for the case of a massive particle moving on an arbitrary curved manifold. The investigation is carried out from the symmetry perspective. By exploiting the parametrization…
The path integral formulation of constrained systems leads to obtain the equations of motion as total differential equations in many variables. If these equations are integrable then one can constuct a valid and a canonical phase space…
Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The…
We introduce and study a novel class of classical integrable many-body systems obtained by generalized $T\bar{T}$-deformations of free particles. Deformation terms are bilinears in densities and currents for the continuum of charges…
We extend the construction of 2D superintegrable Hamiltonians with separation of variables in spherical coordinates using combinations of shift, ladder, and supercharge operators to models involving rational extensions of the two-parameter…
We investigate integrable 2-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of…
We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.
We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.
We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of…
We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail, and…
This is a continuation of the work initiated in a previous paper on so-called driven cofactor systems, which are partially decoupling second-order differential equations of a special kind. The main purpose in that paper was to obtain an…
Quantum integrable systems have very strong mathematical properties that allow an exact description of their energetic spectrum. From the Bethe equations, I formulate the Baxter "T-Q" relation, that is the starting point of two…
Separable coordinate systems are introduced in the complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also…
This is the first part of a two-part paper describing a new concept of separation of variables applied to the Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: 1) Separating coordinates are constructed…
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified…
There is a growing interest in developing covariance functions for processes on the surface of a sphere due to wide availability of data on the globe. Utilizing the one-to-one mapping between the Euclidean distance and the great circle…
We investigate $(d+1)$-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case $d\geq 3$ we formulate a conjecture that any such system with an irreducible dispersion relation must be…