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Physica A Statistical Mechanics and its Applications

Entropy and complexity analysis of Dirac-delta-like quantum potentials

Peter Alexander Bouvrie, Juan Carlos Angulo, J. S. Dehesa

Published Feb 19, 2011• 47 citations
Publisher
DOI
10.1016/j.physa.2011.02.020

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The Dirac delta function δ(x) is widely used in many areas of physics and mathematics. Here we consider the generalization of a Dirac delta function to allow the use of complex arguments. We show that the properties of a generalized delta function are very different from those of a Dirac delta function and that they behave more like a pole in the complex plane. We use the generalized delta function to derive the Glauber-Sudarshan P-function, P(α), for a Schrödinger cat state in a surprisingly simple form. Aside from their potential applications in classical electromagnetism and quantum optics, these results provide insight into the ability of the diagonal P-function to describe density operators with off-diagonal elements.
A curious feature of complex scattering potentials v ( x ) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., , and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v ( x ) = z − δ( x + a ) + z + δ( x − a ), where z ± and a are respectively complex and real parameters and δ( x ) is the Dirac delta function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing -symmetry.
A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., − d2 dx 2 +v(x), and its adjoint. We establish the equivalence of this description with the mathematicians ’ definition of spectral singularities for the potential v(x) = z−δ(x + a) + z+δ(x − a), where z ± and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry. PACS numbers: 03.65.-w