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Renormalized Harmonic-Oscillator Description of Confined Electron Systems with Inverse-Square Interaction  [PDF]
N. Kawakami
Physics , 1993, DOI: 10.1143/JPSJ.62.4163
Abstract: An integrable model for SU($\nu$) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities.
Unitary relation for the time-dependent SU(1,1) systems  [PDF]
Dae-Yup Song
Physics , 2003, DOI: 10.1103/PhysRevA.68.012108
Abstract: The system whose Hamiltonian is a linear combination of the generators of SU(1,1) group with time-dependent coefficients is studied. It is shown that there is a unitary relation between the system and a system whose Hamiltonian is simply proportional to the generator of the compact subgroup of the SU(1,1). The unitary relation is described by the classical solutions of a time-dependent (harmonic) oscillator. Making use of the relation, the wave functions satisfying the Schr\"{o}dinger equation are given for a general unitary representation in terms of the matrix elements of a finite group transformation (Bargmann function). The wave functions of the harmonic oscillator with an inverse-square potential is studied in detail, and it is shown that, through an integral, the model provides a way of deriving the Bargmann function for the representation of positive discrete series of the SU(1,1).
On the inverse to the harmonic oscillator  [PDF]
Marco Cappiello,Luigi Rodino,Joachim Toft
Mathematics , 2013,
Abstract: Let $b_d$ be the Weyl symbol of the inverse to the harmonic oscillator on $\R^d$. We prove that $b_d$ and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for $b_d$. In the even-dimensional case we characterize $b_d$ in terms of elementary functions. In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions.
Exact solution for a harmonic oscillator with a time-dependent inverse square potential by path-integral
具有含时平方反比项的谐振子的路径积分求解

Wang Ping,Yang Xin-E,Song Xiao-Hui,
王平
,杨新娥,宋小会

物理学报 , 2003,
Abstract: Using coordinate-transformation we transformed a harmonic oscillator with time-d ependent mass and a time-dependent inverse potential into a harmonic oscillator with time-independent mass and a time-independent inverse potential accordingly. In terms of the relation between these two different harmonic oscillators' prop agators we derived the exact wavefunction of the former by Feynman path-integral . We also discussed the harmonic oscillator with more additional potentials.
A square root of the harmonic oscillator  [PDF]
R Delbourgo
Physics , 1995, DOI: 10.1103/PhysRevA.52.3356
Abstract: Allowing for the inclusion of the parity operator, it is possible to construct an oscillator model whose Hamiltonian admits an EXACT square root, which is different from the conventional approach based on creation and annihilation operators. We outline such a model, the method of solution and some generalizations.
Collective motions of a quantum gas confined in a harmonic trap  [PDF]
Dae-Yup Song
Physics , 2006, DOI: 10.1103/PhysRevA.72.023614
Abstract: Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency $\nu_c$ is invariant under a squeeze-type unitary transformation, which gives collective {\it radial} breathing motion of frequency $2\nu_c$ to the gas. The amplitudes of the oscillating and breathing motions from the {\it exact} invariances could be arbitrarily large. For a Fermi system, appearance of $2\nu_c$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.
Inverting Time-Dependent Harmonic Oscillator Potential by a Unitary Transformation and a New Class of Exactly Solvable Oscillators  [PDF]
Ali Mostafazadeh
Physics , 1996, DOI: 10.1103/PhysRevA.55.4084
Abstract: A time-dependent unitary (canonical) transformation is found which maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly.
On the realizable classes of the square root of the inverse different in the unitary class group  [PDF]
Cindy Tsang
Mathematics , 2015,
Abstract: Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a finite abelian group of odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ denote its square root of the inverse different, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then the pair $(A_h,Tr_h)$ is locally $G$-isometric to $(\mathcal{O}_KG,t_K)$ and hence defines a class in the unitary class group $\mbox{UCl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. Here $Tr_h$ denotes the trace of $K_h/K$ and $t_K$ the symmetric bilinear form on $\mathcal{O}_KG$ for which $t_K(s,t)=\delta_{st}$ for all $s,t\in G$. We study the collection of all such classes and show that a subset of them is in fact a subgroup of $\mbox{UCl}(\mathcal{O}_KG)$.
The inverse problem for perturbed harmonic oscillator on the half-line  [PDF]
Dmitry Chelkak,Evgeny Korotyaev
Mathematics , 2005,
Abstract: We consider the perturbed harmonic oscillator $T_D\psi=-\psi''+x^2\psi+q(x)\psi$, $\psi(0)=0$ in $L^2(R_+)$, where $q\in H_+=\{q', xq\in L^2(R_+)\}$ is a real-valued potential. We prove that the mapping $q\mapsto{\rm spectral data}={\rm \{eigenvalues of\}T_D{\rm \}}\oplus{\rm \{norming constants\}}$ is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to $q\in H_+$ is given. Moreover, we solve the similar inverse problem for the family of boundary conditions $\psi'(0)=b \psi(0)$, $b\in R$.
Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator  [PDF]
Alexis Pokrovski
Mathematics , 2006, DOI: 10.1007/s11040-007-9025-4
Abstract: Consider quantum harmonic oscillator, perturbed by an even almost-periodic complex-valued potential with bounded derivative and primitive. Suppose that we know the first correction to the spectral asymptotics $\{\Delta\mu_n\}_{n=0}^\infty$ ($\Delta\mu_n=\mu_n-\mu_n^0+o(n^{-1/4})$, where $\mu_n^0$ and $\mu_n$ is the spectrum of the unperturbed and the perturbed operators, respectively). We obtain the formula that recovers the frequencies and the Fourier coefficients of the perturbation.
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