Related papers: Classification of Parameter-Dependent Quantum Inte…
An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(\lambda) \neq [ H^{(N)}(\lambda)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc}…
This paper begins with a review of the well-known KdV hierarchy, the $N$-th Novikov equation, and its finite hierarchy in the classical commutative case. This finite hierarchy consists of $N$ compatible integrable polynomial dynamical…
We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit…
We propose a new unified theoretical framework to construct equivalent representations of the multi-state Hamiltonian operator and present several approaches for the mapping onto the Cartesian phase space. After mapping an F-dimensional…
We propose path integral description for quantum mechanical systems on compact graphs consisting of N segments of the same length. Provided the bulk Hamiltonian is segment-independent, scale-invariant boundary conditions given by…
Quantum computing is among the most far-reaching technologies of the 21st century, tackling challenges at the cutting edge of physics. This new paradigm in computer science harnesses quantum entanglement, one striking non-intuitive feature…
We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are…
In the present paper we examine in a systematic way the most relevant orderings of pure kinetic Hamiltonians for five different position-dependent mass (PDM) profiles: soliton-like, reciprocal quadratic and biquadratic, exponential and…
We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space.…
The linearity inherent in quantum mechanics limits current quantum hardware from directly solving nonlinear systems governed by nonlinear differential equations. One can opt for linearization frameworks such as Carleman linearization, which…
There are several known techniques to construct a Hamiltonian with an expected value that is minimized uniquely by a given quantum state. Common approaches include the parent Hamiltonian construction from matrix product states, building…
A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the…
An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden…
We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more…
Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a…
We analyze a general family of position-dependent mass quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a…
The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are…
Cylindrically symmetric quantum mechanical systems with position dependent masses (PDM) admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them…
We consider a two-spin model, represented classically by a nonlinear autonomous Hamiltonian system with two degrees of freedom and a nontrivial integrability condition, and quantum mechanically by a real symmetric Hamiltonian matrix with…
The construction of exactly-solvable models has recently been advanced by considering integrable $T\bar{T}$ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian…