On the 17th of January 2019, at 10:15 a.m.
A.M.
Ishkhanyan (Ins. Phys. Research NAS of
Armenia)
will give a talk on
"
Solutions of the Schrodinger equations in
terms of the Heun functions"
Abstract
We review
the cases for which the 1D stationary
Schrödinger equation is solved in terms
of the general and (multi-)confluent
Heun functions. We present the possible
choices for coordinate transformation
that provide energy-independent
potentials that are proportional to an
energy-independent continuous parameter
and have a shape independent of that
parameter. In contrast to the
hypergeometric case, no Heun potential
can in general be transformed into
another one by specifications of the
involved parameters.
We show that there exist in total 29
independent Heun potentials. There are
eleven independent potentials that admit
the solution in terms of the general
Heun functions, for nine independent
seven-parametric potentials the solution
is given in terms of the
single-confluent Heun functions, there
are three independent double-confluent
and five independent bi-confluent Heun
potentials (the six-parametric
Lemieux-Bose potentials), and one
tri-confluent Heun potential (the
general five-parametric quartic
oscillator).
There are several independent potentials
that present distinct generalizations of
either a hypergeometric or a confluent
hypergeometric classical potential, some
potentials possess sub-cases of both
hypergeometric types, and others possess
particular conditionally integrable
ordinary or confluent hypergeometric
sub-potentials. We present several
examples of explicit solutions for the
latter potentials.
We show that there exist other exactly
or conditionally integrable
sub-potentials the solution for which is
written in terms of simpler special
functions. However, these are solutions
of different structure. For instance,
there are sub-potentials for which each
of the two fundamental solutions of the
Schrödinger equation is written in terms
of irreducible combinations of
hypergeometric functions. Several such
potentials are derived with the use of
deformed Heun equations. A complementary
approach is the termination of the
hypergeometric series expansions of the
solutions of the Heun equations.
The seminar takes place on Thursdays from 10:15 a.m. to
12:00 in the room 2.23 of the main building of the
Faculty of Physics (the 2nd floor), Pasteur Str. 5,
Warszawa.
Additional information can be found on the webpage http://oldwww.fuw.edu.pl/KMMF/sem.czw.przedp.html.
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