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    Green's function method for single-particle resonant states in relativistic mean field theory

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    Relativistic mean field theory is formulated with the Green's function method in coordinate space to investigate the single-particle bound states and resonant states on the same footing. Taking the density of states for free particle as a reference, the energies and widths of single-particle resonant states are extracted from the density of states without any ambiguity. As an example, the energies and widths for single-neutron resonant states in 120^{120}Sn are compared with those obtained by the scattering phase-shift method, the analytic continuation in the coupling constant approach, the real stabilization method and the complex scaling method. Excellent agreements are found for the energies and widths of single-neutron resonant states.Comment: 20 pages, 7 figure

    Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery

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    This paper establishes a sharp condition on the restricted isometry property (RIP) for both the sparse signal recovery and low-rank matrix recovery. It is shown that if the measurement matrix AA satisfies the RIP condition δkA<1/3\delta_k^A<1/3, then all kk-sparse signals β\beta can be recovered exactly via the constrained 1\ell_1 minimization based on y=Aβy=A\beta. Similarly, if the linear map M\cal M satisfies the RIP condition δrM<1/3\delta_r^{\cal M}<1/3, then all matrices XX of rank at most rr can be recovered exactly via the constrained nuclear norm minimization based on b=M(X)b={\cal M}(X). Furthermore, in both cases it is not possible to do so in general when the condition does not hold. In addition, noisy cases are considered and oracle inequalities are given under the sharp RIP condition.Comment: to appear in Applied and Computational Harmonic Analysis (2012