Sen Gupta, Indranil (2010-08). Analysis of the Three-dimensional Superradiance Problem and Some Generalizations. Doctoral Dissertation. Thesis uri icon

abstract

  • We study the integral equation related to the three and higher dimensional superradiance problem. Collective radiation phenomena has attracted the attention of many physicists and chemists since the pioneering work of R. H. Dicke in 1954. We first consider the three-dimensional superradiance problem and find a differential operator that commutes with the integral operator related to the problem. We find all the eigenfunctions of the differential operator and obtain a complete set of eigensolutions for the three-dimensional superradiance problem. Generalization of the three-dimensional superradiance integral equation is provided. A commuting differential operator is found for this generalized problem. For the three dimensional superradiance problem, an alternative set of complete eigenfunctions is also provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the differential operator commuting with that kernel is indicated. Finally, a concentration problem for the signals which are bandlimited in disjoint frequency-intervals is considered. The problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted in a given time interval. A numerical algorithm for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L2(-?,?) are also proved. Numerical computations are carried out in support of the theory.
  • We study the integral equation related to the three and higher dimensional
    superradiance problem. Collective radiation phenomena has attracted the attention
    of many physicists and chemists since the pioneering work of R. H. Dicke in 1954.
    We first consider the three-dimensional superradiance problem and find a differential
    operator that commutes with the integral operator related to the problem. We
    find all the eigenfunctions of the differential operator and obtain a complete set of
    eigensolutions for the three-dimensional superradiance problem. Generalization of
    the three-dimensional superradiance integral equation is provided. A commuting differential
    operator is found for this generalized problem. For the three dimensional
    superradiance problem, an alternative set of complete eigenfunctions is also provided.
    The kernel for the superradiance problem when restricted to one-dimension is the
    same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the
    differential operator commuting with that kernel is indicated. Finally, a concentration
    problem for the signals which are bandlimited in disjoint frequency-intervals is
    considered. The problem is to determine which bandlimited signals lose the smallest
    fraction of their energy when restricted in a given time interval. A numerical
    algorithm for solution and convergence theorems are given. Orthogonality properties
    of analytically extended eigenfunctions over L2(-?,?) are also proved. Numerical
    computations are carried out in support of the theory.

publication date

  • August 2010