76,032 research outputs found

### On the 2-mode and $k$-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 $\omega$ and coupling strength
$g$ 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 $k\geq 3$ the $k$-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 $k\geq 3$ case is not
diagonalizable, unlike its RWA cousin, the $k$-photon Jaynes-Cummings model
which can be completely diagonalized for all $k$.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: $osp(1|2)$ and its quantum version
$U_t[osp(1|2)]$, $osp(2|2)$ in the non-standard and standard bases and
$gl(2|2)$ 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

### Exact polynomial solutions of second order differential equations and their applications

We find all polynomials $Z(z)$ such that the differential equation
${X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0,$ where $X(z), Y(z),
Z(z)$ are polynomials of degree at most 4, 3, 2 respectively, has polynomial
solutions $S(z)=\prod_{i=1}^n(z-z_i)$ of degree $n$ with distinct roots $z_i$.
We derive a set of $n$ algebraic equations which determine these roots. We also
find all polynomials $Z(z)$ 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 $\phi^6$-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

### Hidden $sl(2)$-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 $sl(2)$-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 $U(1)$ symmetry.Comment: LaTex 14 pages. Version to appear in Annals of Physic

### Community Structure Detection in Complex Networks with Partial Background Information

Constrained clustering has been well-studied in the unsupervised learning
society. However, how to encode constraints into community structure detection,
within complex networks, remains a challenging problem. In this paper, we
propose a semi-supervised learning framework for community structure detection.
This framework implicitly encodes the must-link and cannot-link constraints by
modifying the adjacency matrix of network, which can also be regarded as
de-noising the consensus matrix of community structures. Our proposed method
gives consideration to both the topology and the functions (background
information) of complex network, which enhances the interpretability of the
results. The comparisons performed on both the synthetic benchmarks and the
real-world networks show that the proposed framework can significantly improve
the community detection performance with few constraints, which makes it an
attractive methodology in the analysis of complex networks

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