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Register a new userOn the one-dimensional harmonic oscillator with a singular perturbation
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In this paper we investigate the one-dimensional harmonic oscillator with a singular perturbation concentrated in one point. We describe all possible selfadjoint realizations and we show that for certain conditions on the perturbation exactly one negative eigenvalues can arise. This eigenvalue tends to $-infty$ as the perturbation becomes stronger.
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Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrodinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU(1,1) group-related coherent states for the SO minimize the Robertson and Schrodinger relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.
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.
The time operator for a quantum singular oscillator of the Calogero-Sutherland type is constructed in terms of the generators of the SU(1,1) group. In the space spanned by the eigenstates of the Hamiltonian, the time operator is not self-adjoint. We show, that the time-energy uncertainty relation can be given the meaning within the Barut-Girardello coherent states defined for the singular oscillator.We have also shown the relationship with the time-of-arrival operator of Aharonov and Bohm.
A new recursion procedure for deriving renormalized perturbation expansions for the one-dimensional anharmonic oscillator is offered. Based upon the $hbar$-expansions and suitable quantization conditions, the recursion formulae obtained have the same simple form both for ground and excited states and can be easily applied to any renormalization scheme. As an example, the renormalized expansions for the sextic anharmonic oscillator are considered.
In this paper we prove Hormander-Mihlin multiplier theorems for pseudo-multipliers associated to the harmonic oscillator (also called the Hermite operator). Our approach can be extended to also obtain the $L^p$-boundedness results for multilinear pseudo-multipliers. By using the Littlewood-Paley theorem associated to the harmonic oscillator we also give $L^p$-boundedness and $L^p$-compactness properties for multipliers. $(L^p,L^q)$-estimates for spectral pseudo-multipliers also are investigated.
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