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Register a new userThe two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation
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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 compute 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 $lambda$.
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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.
We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic function for example). It is shown that the eigenvalues of H+V have asymptotics of the form lambda_n(H+V)=lambda_n(H)+W(sqrt n)n^{-1/4}+O(n^{-1/2}ln(n)) as nto+infty, where W is a quasi-periodic function which can be defined explicitly in terms of V.
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.
Quasi-periodically driven quantum parametric oscillators have been the subject of several recent investigations. Here we show that for such oscillators, the instability zones of the mean position and variance (alternatively the mean energy) for a time developing wave packet are identical for the strongest resonance in the three-dimensional parameter space of the quasi-periodic modulation as it is for the two-dimensional parameter space of the periodic modulations.
Let $H$ denote the harmonic oscillator Hamiltonian on $mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrodinger propagator $U(t)=e^{-itH},$ and find that while $operatorname{singsupp} operatorname{Tr} U(t) subset 2 pi mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d geq 2$ for which the singularities of $operatorname{Tr} U(t)$ at nonzero multiples of $2 pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(lambda^{d-1})$, improving in these cases the $O(lambda^{d-1})$ remainder previously established by Helffer--Robert.
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