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تسجيل مستخدم جديدQuantization of the Damped Harmonic Oscillator Revisited
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We return to the description of the damped harmonic oscillator by means of a closed quantum theory with a general assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model recently proposed by one of the authors. We show the local equivalence between the two models and argue that latter has better high energy behavior and is naturally connected to existing open-quantum-systems approaches.
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In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to c
ompute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wangs results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling constant $lambda$.
This Letter is based on the $kappa$-Dirac equation, derived from the $kappa$-Poincar{e}-Hopf algebra. It is shown that the $kappa$-Dirac equation preserves parity while breaks charge conjugation and time reversal symmetries. Introducing the Dirac osc
illator prescription, $mathbf{p}tomathbf{p}-imomegabetamathbf{r}$, in the $kappa$-Dirac equation, one obtains the $kappa$-Dirac oscillator. Using a decomposition in terms of spin angular functions, one achieves the deformed radial equations, with the associated deformed energy eigenvalues and eigenfunctions. The deformation parameter breaks the infinite degeneracy of the Dirac oscillator. In the case where $varepsilon=0$, one recovers the energy eigenvalues and eigenfunctions of the Dirac oscillator.
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I have made an ample study of one dimensional quantum oscillators, ranging from logarithmic to exponential potentials. I have found that the eigenvalues of the hamiltonian of the oscillator with the limiting (approachissimo) harmonic potential (~ p(x
)2) maps the zeros of the Riemann function height up in the Riemann line. This is the potential created by the field of J(x) that is the Riemann generator of the prime number counting function, p(x), that in turn can be defined by an integral transformation of the Riemann zeta function. This plays the role of the spring strength of the quantum limiting harmonic oscillator. The number theory meaning of this result is that the roots height up of the zeta function are the eigenvalues of a Hamiltonian whose potential is the number of primes squared up to a given x. Therefore this may prove the never published Hilbert-Polya conjecture. The conjecture is true but does not imply the truth of the Riemann hypothesis. We can have complex conjugated zeros off the Riemman line and map them with another hermitic operator and a general expression is given for that. The zeros off the line affect the fluctuation of the eigenvalues but not their mean values.
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