The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this curious property. These models have the notable feature that they do not preserve U(1) symmetry, which is typically associated to a non-conservation of particle number. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtained via the algebraic Bethe ansatz formalism.
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``
XXX spin models --- quasi-
XXX spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra
A^1, defined as the real subalgebra of the Weyl algebra
A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for
A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in
A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in
A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in
A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of
A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in
A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.