相关论文: Operator Transformations Between Exactly Solvable …
Effective interaction operators usually act on a restricted model space and give the same energies (for Hamiltonian) and matrix elements (for transition operators etc.) as those of the original operators between the corresponding true…
We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $…
We show that similarity (or equivalent) transformations enable one to construct non-Hermitian operators with real spectrum. In this way we can also prove and generalize the results obtained by other authors by means of a gauge-like…
A periodic linear graph operator acts on states (functions) defined on the vertices of a graph equipped with a free translation action. Fourier transform with respect to the translation group reveals the central spectral objects, Bloch and…
The exact bound state spectrum of rationally extended shape invariant real as well as $PT$ symmetric complex potentials are obtained by using potential group approach. The generators of the potential groups are modified by introducing a new…
Examples are given of non-Hermitian Hamiltonian operators which have a real spectrum. Some of the investigated operators are expressed in terms of the generators of the Weil-Heisenberg algebra. It is argued that the existence of an…
Complete sets of bases of differential invariants, operators of invariant differentiation and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of…
In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum…
An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.
Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schr\"odinger equation are considered. Due to the fact that…
We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators…
For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly.
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…
We deform the real potential of Poeschl and Teller by a shift of its coordinate in imaginary direction. We show that the new model remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and…
We introduce the concepts of the Fourier transform and convolution generated by an arbitrary restriction of the differentiation operator in the space $L_{2}(0,b).$ In contrast to the classical convolution, the introduced convolution…
Most of the known Fourier transforms associated with the equations of mathematical physics have a trivial kernel, and an inversion formula as well as the Parseval equality are fulfilled. In other words, the system of the eigenfunctions…
We use the method of similar operators to study a mixed problem for a differential equation with an involution and an operator-valued potential function. The differential operator defined by the equation is transformed into a similar…
Different cases of sequences of the Laplace Transformations for the 2D Schrodinger operator in the periodic magnetic field and electric potential are considered. They lead to the exactly solvable operators with nonstandard spectral…
A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials.…
Under some hypotheses (symmetry, confluence), we enumerate all quadratically presented algebras, generated by creation and destruction operators, in which number operators exist. We show that these are algebras of bosons, fermions, their…