9,808 research outputs found
Exact polynomial solutions of second order differential equations and their applications
We find all polynomials such that the differential equation
where are polynomials of degree at most 4, 3, 2 respectively, has polynomial
solutions of degree with distinct roots .
We derive a set of algebraic equations which determine these roots. We also
find all polynomials which give polynomial solutions to the differential
equation when the coefficients of X(z) and Y(z) are algebraically dependent. As
applications to our general results, we obtain the exact (closed-form)
solutions of the Schr\"odinger type differential equations describing: 1) Two
Coulombically repelling electrons on a sphere; 2) Schr\"odinger equation from
kink stability analysis of -type field theory; 3) Static perturbations
for the non-extremal Reissner-Nordstr\"om solution; 4) Planar Dirac electron in
Coulomb and magnetic fields; and 5) O(N) invariant decatic anharmonic
oscillator.Comment: LaTex 25 page
On the 2-mode and -photon quantum Rabi models
By mapping the Hamiltonians of the two-mode and 2-photon Rabi models to
differential operators in suitable Hilbert spaces of entire functions, we prove
that the two models possess entire and normalizable wavefunctions in the
Bargmann-Hilbert spaces only if the frequency and coupling strength
satisfy certain constraints. This is in sharp contrast to the quantum Rabi
model for which entire wavefunctions always exist. For model parameters
fulfilling the aforesaid constraints we determine transcendental equations
whose roots give the regular energy eigenvalues of the models. Furthermore, we
show that for the -photon Rabi model does not possess
wavefunctions which are elements of the Bargmann-Hilbert space for all
non-trivial model parameters. This implies that the case is not
diagonalizable, unlike its RWA cousin, the -photon Jaynes-Cummings model
which can be completely diagonalized for all .Comment: LaTex 15 pages. Version to appear in Reviews in Mathematical Physic
Super Coherent States, Boson-Fermion Realizations and Representations of Superalgebras
Super coherent states are useful in the explicit construction of
representations of superalgebras and quantum superalgebras. In this
contribution, we describe how they are used to construct (quantum)
boson-fermion realizations and representations of (quantum) superalgebras. We
work through a few examples: and its quantum version
, in the non-standard and standard bases and
in the non-standard basis. We obtain free boson-fermion realizations
of these superalgebras. Applying the boson-fermion realizations, we explicitly
construct their finite-dimensional representations. Our results are expected to
be useful in the study of current superalgebras and their corresponding
conformal field theories.Comment: LaTex 20 pages. Invited contribution for the volume "Trends in Field
Theory Research" by Nova Science Publishers Inc., New York, 2004. Accepted
for publication in the volum
Hidden -algebraic structure in Rabi model and its 2-photon and two-mode generalizations
It is shown that the (driven) quantum Rabi model and its 2-photon and 2-mode
generalizations possess a hidden -algebraic structure which explains the
origin of the quasi-exact solvability of these models. It manifests the first
appearance of a hidden algebraic structure in quantum spin-boson systems
without symmetry.Comment: LaTex 14 pages. Version to appear in Annals of Physic
Relationship between Nichols braided Lie algebras and Nichols algebras
We establish the relationship among Nichols algebras, Nichols braided Lie
algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra
is finite-dimensional if and only if Nichols braided Lie
algebra is finite-dimensional if there does not exist any
-infinity element in ; (ii) Nichols Lie algebra is infinite dimensional if is infinite. We give the sufficient
conditions for Nichols braided Lie algebra to be a homomorphic
image of a braided Lie algebra generated by with defining relations.Comment: LeTex 18 pages, need JOLT-macros to compile. To appear in Journal of
Lie Theor
One loop amplitude from null string
We generalize the CHY formalism to one-loop level, based on the framework of
the null string theory. The null string, a tensionless string theory, produces
the same results as the ones from the chiral ambitwistor string theory, with
the latter believed to give a string interpretation of the CHY formalism. A key
feature of our formalism is the interpretation of the modular parameters. We
find that the modular transformation invariance of the ordinary string
theory does not survive in the case of the null string theory. Treating the
integration over the modular parameters this way enable us to derive the n-gons
scattering amplitude in field theory, thus proving the n-gons conjecture.Comment: 18 pages, 2 figure
On Nichols (braided) Lie algebras
We prove {\rm (i)} Nichols algebra of vector space is
finite-dimensional if and only if Nichols braided Lie algebra
is finite-dimensional; {\rm (ii)} If the rank of connected is and
is an arithmetic root system, then and {\rm (iii)} if is an arithmetic
root system and there does not exist any -infinity element with for any , then if and
only if there exists , which is twisting equivalent to , such that Furthermore we give an estimation of
dimensions of Nichols Lie algebras and two examples of Lie algebras which do
not have maximal solvable ideals.Comment: 29 Pages; Substantially revised version; To appear in International
Journal of Mathematic
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