A remarkable spectral feature of the Schrödinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ'-interaction centred at the origin: double degeneracy and level crossing (original) (raw)
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Reports on Mathematical Physics, 2012
In this paper the self-adjoint Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identical attractive point interactions (delta distributions) situated symmetrically with respect to the equilibrium position of the oscillator is rigorously defined by means of its resolvent (Green's function). The equations determining the even and odd eigenvalues of the Hamiltonian are explicitly provided in order to shed light on the behaviour of such energy levels both with respect to the separation distance between the point interaction centres and to the coupling constant.
arXiv (Cornell University), 2017
We decorate the one-dimensional conic oscillator 1 2 − d 2 dx 2 + |x| with a point impurity of either δ-type, or local δ-type or even nonlocal δ-type. All the three cases are exactly solvable models, which are explicitly solved and analysed, as a first step towards higher dimensional models of physical relevance. We analyse the behaviour of the change in the energy levels when an interaction of the type −λ δ(x) or −λ δ(x − x 0) is switched on. In the first case, even energy levels (pertaining to antisymmetric bound states) remain invariant with λ although odd energy levels (pertaining to symmetric bound states) decrease as λ increases. In the second, all energy levels decrease when the form factor λ increases. A similar study has been performed for the so called nonlocal δ interaction, requiring a coupling constant renormalization, which implies the replacement of the form factor λ by a renormalized form factor β. In terms of β, even energy levels are unchanged. However, we show the existence of level crossings: after a fixed value of β the energy of each odd level, with the natural exception of the first one, becomes lower than the constant energy of the previous even level. Finally, we consider an interaction of the type −aδ(x) + bδ (x), and analyse in detail the discrete spectrum of the resulting self-adjoint Hamiltonian.
Perturbation theory of odd anharmonic oscillators
1980
We study the perturbation theory for H = p 2 + x 2 +βx 2n+ί , n=l,2,.... It is proved that when ImβΦO, H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schrodinger perturbation expansion, and admits an analytic continuation to Imβ = 0 where it can be interpreted as a resonance of the problem.
The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation
The European Physical Journal Plus
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 Wang's 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 λ.
On eigenproblem for inverted harmonic oscillators
arXiv (Cornell University), 2019
We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in C ∞ (R). The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is C. The spectrum of the differential operator − d dx 2 − ω 2 x 2 is continuous and has physical significance only for the states which are in L 2 (R) and correspond to real eigenvalues. To identify them we orthonormalize in Dirac sense the states corresponding to real eigenvalues. This leads to the doubly degenerated real line as the spectrum of the Hamiltonian (in L 2 (R)). We also use two other approaches. First we define a unitary operator between L 2 (R) and L 2 for two copies of R. This operator has the property that the spectrum of the image of the inverted harmonic oscillator corresponds to the spectrum of the operator −i d dx. This shows again that the (generalized) spectrum of the inverted harmonic operator is a doubly degenerated real line. The second approach uses rigged Hilbert spaces.