**by Sergei K. Suslov**

School of Mathematics and Statistics
Sciences&Mathematical, Computational
and Modeling Sciences Center

Arizona State University

Tempe, AZ 85287-1804

USA

*Title**:***On the Harmonic Oscillator Group**

*Authors*: Raquel M. Lopez, Sergei K. Suslov and Jose Vega-Guzman

*Comments:* 11 pages, no figures; for some *Mathematica* proofs
and animations, please see BerrySummary HarmonicOscillatorGroup.nb

(For *Mathematica* proofs of Lemmas 1-3, please see: Koutschan.nb
)*
Abstract:* We discuss the maximum kinematical invariance group of the
quantum harmonic oscillator from a viewpoint of the Ermakov-type system.
A six parameter family of the square integrable oscillator wave functions,
which seems cannot be obtained by the standard separation of variables, is
presented as an example. The invariance group of generalized driven harmonic
oscillator is shown to be isomorphic to the corresponding Schroedinger group
of the free particle.

https://arxiv.org/abs/1111.5569

*Title**:***Dynamical Invariants and Berry's
Phase for Generalized Driven Harmonic Oscillators**

*Authors*: Barbara Sanborn, Sergei K. Suslov and Luc Vinet

*Comments:* 10 pages, no figures; published in Journal of Russian Laser
Research, Volume 32, Issue 5 (2011), Pages 486-494; https://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10946-011-9238-7

For some *Mathematica* details, please see BerrySummary

*Abstract:* We present quadratic dynamical invariant and evaluate Berry's
phase for the time-dependent Schroedinger equation with the most general
variable quadratic Hamiltonian.

https://arxiv.org/abs/1108.5144

*Title:* **Exact Wave Functions for Generalized Harmonic Oscillators
**

https://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10946-011-9223-1.

https://arxiv.org/abs/1102.5119

*Title:* **The Riccati System and a Diffusion-Type Equation**

*Authors:* Erwin Suazo, Sergei K. Suslov, and Jose M.
Vega-Guzman

*Comments:* 11 pages, no figures

*Abstract:* We discuss a method of constructing solution of the initial
value problem for duffusion-type equations in terms of solutions of certain
Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation
is also considered.

https://arxiv.org/abs/1102.4630

*Title:* **The Riccati Differential Equation and a Diffusion-Type
Equation**

*Authors:* Erwin Suazo, Sergei K. Suslov, and Jose M.
Vega-Guzman

*Comments:* 12 pages, no figures

*Abstract:* We construct an explicit solution of the
Cauchy initial value problem for certain diffusion-type equations
with variable coefficients on the entire real line. The corresponding
Green function (heat kernel) is given in terms of elementary functions
and certain integrals involving a characteristic function, which should
be found as an analytic or numerical solution of the second order linear
differential equation with time-dependent coefficients. Some special
and limiting cases are outlined. Solution of the corresponding non-homogeneous
equation is also found.

Published in New York Journal of Mathematics, Vol. 17a (2011):
https://nyjm.albany.edu:8000/j/2011/Vol17a.htm

https://nyjm.albany.edu:8000/j/2011/17a-14.html

https://arxiv.org/abs/0807.4349

*Title:* **A Graphical Approach to a Model of Neuronal Tree with
Variable Diameter**

*Authors:* Marco Herrera-Valdez and Sergei K. Suslov

*Comments:* 14 pages, 3 figures

*Abstract:* We propose a simple graphical approach to steady state
solutions of the cable equation for a general model of dendritic tree with
tapering. A simple case of transient solutions is also briefly discussed.

https://arxiv.org/abs/1101.0296

*Title:* **On Integrability of Nonautonomous Nonlinear Schroedinger
Equations**

*Author:* Sergei K. Suslov

*Comments:* 11 pages, no figures.

*Abstract:* We show, in general, how to transform nonautonomous
nonlinear Schroedinger equation with quadratic Hamiltonians into the standard
autonomous form that is completely integrable by the familiar inverse scattering
method in nonlinear science. Derivation of the corresponding equivalent
nonisospectral Lax pair is outlined.

https://arxiv.org/abs/1012.3661

*Title:* **Soliton-like Solutions for Nonlinear Schroedinger Equation
with Variable Quadratic Hamiltonians**

*Authors:* Erwin Suazo and Sergei K. Suslov

*Comments:* 22 pages, 1 figure

*Abstract:* We construct one soliton solutions for the
nonlinear Schroedinger equation with variable quadratic Hamiltonians in
a unified form by taking advantage of a complete (super) integrability of
generalized harmonic oscillators. The soliton wave evolution in external
fields with variable quadratic potentials is totally determined by the linear
problem, like motion of a classical particle with acceleration, and the (self-similar)
soliton shape is due to a subtle balance between the linear Hamiltonian
(dispersion and potential) and nonlinearity in the Schroedinger equation
by the standards of soliton theory. Most linear (hypergeometric, Bessel)
and a few nonlinear (Jacobian elliptic, second Painleve transcendental)
classical special functions of mathematical physics are linked together
through these solutions, thus providing a variety of nonlinear integrable
cases. Examples include bright and dark solitons, and Jacobi elliptic and
second Painleve transcendental solutions for several variable Hamiltonians
that are important for current research in nonlinear optics and Bose-Einstein
condensation. The Feshbach resonance matter wave soliton management is
briefly discussed from this new perspective.

https://arxiv.org/abs/1010.2504

https://arxiv4.library.cornell.edu/abs/1008.2534v1

Title

https://arxiv.org/abs/1006.3362

*Title:* **Dynamical Invariants for Variable Quadratic Hamiltonians**

*Author:* Sergei K. Suslov

*Comments:* 21 pages, no figures, published in Physica
Scripta:

https://iopscience.iop.org/1402-4896/81/5/055006

*Abstract:* We consider linear and quadratic integrals of motion
for general variable quadratic Hamiltonians. Fundamental relations between
the eigenvalue problem for linear dynamic invariants and solutions of
the corresponding Cauchy initial value problem for the time-dependent
Schroedinger equation are emphasized. An eigenfunction expansion of the
solution of the initial value problem is also found. A nonlinear superposition
principle for generalized Ermakov systems is established as a result
of decomposition of the general quadratic invariant in terms of the linear
ones. https://arxiv.org/abs/1002.0144

*Title:* **Quantum Integrals of Motion for Variable Quadratic Hamiltonians**

*Authors:* Ricardo Cordero-Soto, Erwin Suazo, and Sergei
K. Suslov

*Comments:* 32 pages, no figures:
Published in Annals of Physics

*Abstract:* We construct the integrals of motion for several models
of the quantum damped oscillators in nonrelativistic quantum mechanics
in a framework of a general approach to the time-dependent Schroedinger
equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld
dynamical invariant is given. The time-evolution of the expectation
values of the energy related positive operators is determined for the
oscillators under consideration. A proof of uniqueness of the corresponding
Cauchy initial value problem is discussed as an application. https://arxiv.org/abs/0912.4900

*Title:* **Mathematical Structure of Relativistic
Coulomb Integrals***
Author:* S. K. Suslov

Comments:

Abstract:

https://www.springerlink.com/content/93468l2q34218753/

https://iopscience.iop.org/0953-4075/43/7/074006

https://arxiv.org/abs/0908.3021

https://arxiv.org/abs/0907.1609

https://arxiv.org/abs/0908.0032

https://stacks.iop.org/0953-4075/42/185003

Abstract: We evaluate the matrix elements <Or^{p}>, where O ={1, \beta, i\alpha n \beta} are the standard Dirac matrix operators for the relativistic Coulomb problems, in terms of the generalized hypergeometric functions_{3}F_{2} for all suitable powers. Their connections with the Chebyshev and Hahn polynomials of a discrete variable are emphasized. As a result, we derive two sets of Pasternack-type matrix identities for these integrals, when p->-p-1 and p->-p-3, respectively. Some applications to the theory of hydrogenlike relativistic systems are reviewed.

https://arxiv.org/abs/0906.3338

https://www.ashdin.com/journals/jpm/2009/S090603.htm

https://arxiv.org/abs/0905.0507

https://arxiv.org/abs/0903.3608

https://83.149.209.141/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=6475&option_lang=eng

https://arxiv.org/abs/0808.3149

Abstract: We discuss an integral form of the Cauchy initial value problem for the nonlinear Schroedinger equation with variable coefficients. Some special and limiting cases are outlined.

https://arxiv.org/abs/0805.0633

https://link.aip.org/link/?JMP/49/072102.

https://www.springerlink.com/content/h523m0u4m378l266/fulltext.pdf

https://link.aip.org/link/?JMAPAQ/49/012104/1

Please see our paper among

https://m.jmp.aip.org/features/most_downloaded?month=2&year=2008

K. Ey, A. Ruffing, and S. K. Suslov,

The Ramanujan Journal, Askey Special Issues, Volume 13, Number 1-3, June 2007, pp. 1-469 (G. E. Andrews, G. Gasper, and S. K. Suslov, Coordinating Editors).

S. K. Suslov,

S. K. Suslov,

R. M. Lopez and S. K. Suslov,

https://arxiv.org/abs/0707.1902

1. A.F. Nikiforov, S.K. Suslov, and V.B. Uvarov,

2. R. Askey, M. Rahman, and S.K. Suslov,

3. S.K. Suslov,

4. R. Cordero-Soto, R.M. Lopez, E. Suazo and S.K. Suslov,

5. S.K. Suslov,

Your comments are welcome: sks@asu.edu