76,032 research outputs found

    On the 2-mode and kk-photon quantum Rabi models

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    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 ω\omega and coupling strength gg 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 k3k\geq 3 the kk-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 k3k\geq 3 case is not diagonalizable, unlike its RWA cousin, the kk-photon Jaynes-Cummings model which can be completely diagonalized for all kk.Comment: LaTex 15 pages. Version to appear in Reviews in Mathematical Physic

    Super Coherent States, Boson-Fermion Realizations and Representations of Superalgebras

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    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(12)osp(1|2) and its quantum version Ut[osp(12)]U_t[osp(1|2)], osp(22)osp(2|2) in the non-standard and standard bases and gl(22)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

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    We find all polynomials Z(z)Z(z) such that the differential equation X(z)d2dz2+Y(z)ddz+Z(z)S(z)=0,{X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0, where X(z),Y(z),Z(z)X(z), Y(z), Z(z) are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions S(z)=i=1n(zzi)S(z)=\prod_{i=1}^n(z-z_i) of degree nn with distinct roots ziz_i. We derive a set of nn algebraic equations which determine these roots. We also find all polynomials Z(z)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 ϕ6\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)sl(2)-algebraic structure in Rabi model and its 2-photon and two-mode generalizations

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    It is shown that the (driven) quantum Rabi model and its 2-photon and 2-mode generalizations possess a hidden sl(2)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)U(1) symmetry.Comment: LaTex 14 pages. Version to appear in Annals of Physic

    Community Structure Detection in Complex Networks with Partial Background Information

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    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