Supersymmetrical intertwining relations of second order in derivatives allow to construct a two-dimensional quantum model with complex potential, for which {\it all} energy levels and bound state wave functions are obtained analytically. This model {\it is not amenable} to separation of variables, and it can be considered as a specific complexified version of generalized two-dimensional Morse model with additional
sinh−2 term. The energy spectrum of the model is proved to be purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. The symmetry operator is found, the biorthogonal basis is described, and the pseudo-Hermiticity of the model is demonstrated. The obtained wave functions are found to be common eigenfunctions both of the Hamiltonian and of the symmetry operator.
A shape invariant nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of exhibiting its hidden algebraic structure. The two operators A+ and A−, coming from the shape invariant supersymmetrical approach, where A+ acts as a raising operator while A− annihilates all wavefunctions, are completed by introducing a novel pair of operators B+ and B−, where B− acts as the missing lowering operator. These four operators then serve as building blocks for constructing gl(2) generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an sp(4) algebra, as well as an osp(1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.
A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze.However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied.We present a set of six operators {A ± , B ± , C ± } that can be combined to build a gl(3) hidden algebra.The latter can be embedded in an sp(6) algebra, as well as in an osp(1/6) superalgebra.The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state.We present the action of these operators and study the construction of an extended biorthogonal basis.These rely on establishing various nontrivial polynomial and commutator identities.We also make a connection between the hidden symmetry and the underlying superintegrability property of the model.Interestingly, the integrals generate a cubic algebra.This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.
We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions as a λ−deformation of the Dunkl oscillator. This deformation is interpreted as the introduction of a non-constant curvature on the underlying space or, equivalently, as a quadratic position-dependent mass for the Dunkl oscillator. This new ND quantum model is shown to be exactly solvable, and its eigenvalues and eigenfunctions are explicitly presented. It is shown that in the 2D case both Darboux III and Dunkl oscillators can be coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of the λ−deformation and of the Dunkl derivatives on the Landau levels can be studied. Finally, the full 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the λ−deformation and the magnetic field is explicitly illustrated.
We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an
sl(2) algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra
U(sl(3)). We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the
sl(2) algebraization after parabolic separation, and the
sl(3) algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common
sl(3) structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.
An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both momentum and position configuration spaces. It is shown that they are expressed through continuous q-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces.