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Register a new userAsymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian
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Dorje C. Brody Professor
Publication date
2017
fields
Physics
and research's language is
English
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The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.
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One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 {it Commun. Pure Appl. Math.} tb{13} 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that $eta$ operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of the spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point the equivalent Hermitian Hamiltonian becomes undetermined.
We introduce generaliz
In this paper, whose aims are mainly pedagogical, we illustrate how to use the local zeta regularization to compute the stress-energy tensor of the Casimir effect. Our attention is devoted to the case of a neutral, massless scalar field in flat space-time, on a space domain with suitable (e.g., Dirichlet) boundary conditions. After a simple outline of the local zeta method, we exemplify it in the typical case of a field between two parallel plates, or outside them. The results are shown to agree with the ones obtained by more popular methods, such as point splitting regularization. In comparison with these alternative methods, local zeta regularization has the advantage to give directly finite results via analitic continuation, with no need to remove or subtract divergent quantities.
We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We show also how the application of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quantum superintegrability of the corresponding quantum Hamiltonian operator.Our results are obtained by applying the theory of extended Hamiltonian systems, which is strictly connected with the geometry of warped manifolds.
In this paper a geometric method based on Grassmann manifolds and matrix Riccati equations to make hermitian matrices diagonal is presented. We call it Riccati Diagonalization.
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