49 research outputs found
Intertwining technique for a system of difference Schroedinger equations and new exactly solvable multichannel potentials
The intertwining operator technique is applied to difference Schroedinger
equations with operator-valued coefficients. It is shown that these equations
appear naturally when a discrete basis is used for solving a multichannel
Schroedinger equation. New families of exactly solvable multichannel
Hamiltonians are found
Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
Diagonalization of a certain operator in irreducible representations of the
positive discrete series of the quantum algebra U_q(su(1,1)) is studied.
Spectrum and eigenfunctions of this operator are found in an explicit form.
These eigenfunctions, when normalized, constitute an orthonormal basis in the
representation space. The initial U_q(su(1,1))-basis and the basis of
eigenfunctions are interrelated by a matrix with entries, expressed in terms of
big q-Laguerre polynomials. The unitarity of this connection matrix leads to an
orthogonal system of functions, which are dual with respect to big q-Laguerre
polynomials. This system of functions consists of two separate sets of
functions, which can be expressed in terms of q-Meixner polynomials
M_n(x;b,c;q) either with positive or negative values of the parameter b. The
orthogonality property of these two sets of functions follows directly from the
unitarity of the connection matrix. As a consequence, one obtains an
orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A
biorthogonal system of functions (with respect to the scalar product in the
representation space) is also derived.Comment: 15 pages, LaTe
Discrete quantum model of the harmonic oscillator
We construct a new model of the quantum oscillator, whose energy spectrum is
equally-spaced and lower-bounded, whereas the spectra of position and momentum
are a denumerable non-degenerate set of points in [-1,1] that depends on the
deformation parameter q from (0,1). We provide its explicit wavefunctions, both
in position and momentum representations, in terms of the discrete q-Hermite
polynomials. We build a Hilbert space with a unique measure, where an analogue
of the fractional Fourier transform is defined in order to govern the time
evolution of this discrete oscillator. In the limit q to 1, one recovers the
ordinary quantum harmonic oscillator.Comment: 21 page
Boundary relations and generalized resolvents of symmetric operators
The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint
exit space extensions of a, not necessarily densely defined, symmetric
operator, in terms of maximal dissipative (in \dC_+) holomorphic linear
relations on the parameter space (the so-called Nevanlinna families). The new
notion of a boundary relation makes it possible to interpret these parameter
families as Weyl families of boundary relations and to establish a simple
coupling method to construct the generalized resolvents from the given
parameter family. The general version of the coupling method is introduced and
the role of boundary relations and their Weyl families for the
Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page
Eleven papers on differential equations
The papers in this volume, like those in the previous one, have been selected, translated, and edited from publications not otherwise translated into English under the auspices of the AMS-ASL-IMS Committee on Translations from Russian and Other Foreign Languages