On the Spectrum of the Schrdinger Hamiltonian of the One-Dimensional Harmonic Oscillator Perturbed by Two Identical Attractive Point Interactions

Fassari, Silvestro; Rinaldi, Fabio

doi:10.1016/s0034-4877(13)60004-x

Cited by 35 publications

(71 citation statements)

References 20 publications

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“…By citing [1] it is important to recall that 'as is well known to Quantum Chemistry students, three-dimensional interactions with a nonzero range do not manifest this singular behavior in the limit of the distance between the two centers shrinking to zero, as the classical textbook example of H + 2 smoothly approaching He + in the limit R → 0 + clearly shows' (see [2][3][4]). The same phenomenon had been observed in [5] (see also [6][7][8][9][10]) dealing with another model involving a pair of identical point interactions symmetrically situated around the origin defined rigorously by using a 'coupling constant renormalization' as well, namely the one-dimensional energy operator in which the kinetic component is given by the Salpeter free Hamiltonian p 2 + m 2 , m > 0.…”

Section: Introductionsupporting

confidence: 62%

“…As can be guessed, the proof is bound to be rather similar to the one of Theorem 2.2(b) in [10] for the one-dimensional analog of the model since the behavior of the following series will play a crucial role:…”

Section: ) Namely the One Of The Self-adjoint Hamiltonian Of The Thmentioning

confidence: 83%

“…By means of implicit differentiation, we can write for any E < 3/2: 10) having simplified the factor 1/π 3/2 . After computing the two partial derivatives (by moving the derivatives inside the integrals, using dominated convergence) we get:…”

Section: Making Sense Of the Merely Formal Expressionmentioning

confidence: 99%

“…As was fully proved in [10], this singular behavior does not occur in the one-dimensional analog of the model given that the Dirac distribution is an infinitesimally small perturbation of the Laplacian in one dimension, which implies that the renormalization procedure is not required at all in that case to define a self-adjoint Hamiltonian (obtained instead by means of the KLMN theorem, see [11]). …”

Section: Introductionmentioning

confidence: 97%

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Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II

Albeverio¹,

Fassari²,

Rinaldi³

2016

Nanosystems: Phys. Chem. Math.

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In this note, we continue our analysis (started in [1]) of the isotropic three-dimensional harmonic oscillator perturbed by a pair of identical attractive point interactions symmetrically situated with respect to the origin, that is to say, the mathematical model describing a symmetric quantum dot with a pair of point impurities. In particular, by making the coupling constant (to be renormalized) dependent also upon the separation distance between the two impurities, we prove that it is possible to rigorously define the unique self-adjoint Hamiltonian that, differently from the one introduced in [1], behaves smoothly as the separation distance between the impurities shrinks to zero. In fact, we rigorously prove that the Hamiltonian introduced in this note converges in the norm-resolvent sense to that of the isotropic three-dimensional harmonic oscillator perturbed by a single attractive point interaction situated at the origin having double strength, thus making this threedimensional model more similar to its one-dimensional analog (not requiring the renormalization procedure) as well as to the three-dimensional model involving impurities given by potentials whose range may even be physically very short but different from zero. Moreover, we show the manifestation of the Zeldovich effect, known also as level rearrangement, in the model investigated herewith. More precisely, we take advantage of our renormalization procedure to demonstrate the possibility of using the concept of 'Zeldovich spiral', introduced in the case of perturbations given by rapidly decaying potentials, also in the case of point perturbations.

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Section: Introductionsupporting

confidence: 62%

Section: ) Namely the One Of The Self-adjoint Hamiltonian Of The Thmentioning

confidence: 83%

Section: Making Sense Of the Merely Formal Expressionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II

Albeverio¹,

Fassari²,

Rinaldi³

2016

Nanosystems: Phys. Chem. Math.

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“…In this work, we wish to further extend our previous research on various types of point perturbations of Schrödinger Hamiltonians with or without harmonic confinement (see, e.g., [2][3][4][5][6][7][8][9][10][11]) to the physical models utilized to describe three-dimensional quantum dots.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin

Albeverio¹,

Fassari²,

Rinaldi³

2016

Nanosystems: Phys. Chem. Math.

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In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive δ-interaction centered at the origin or by a pair of identical attractive δ-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Brüning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ -interaction in one dimension, fully investigated in [2,3], given the existence in both models of the remarkable spectral phenomenon called "level crossing". The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green's function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of "positional disorder", introduced in [1] in the case of a quantum dot with a single δ-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single δ-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive δ-interactions symmetrically situated at the same distance from the origin.Spectral properties of a symmetric three-dimensional quantum dot . . . 269

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The Spectrum of a Harmonic Oscillator Operator Perturbed by $${\delta}$$ δ -Interactions

Mityagin

2016

Integr. Equ. Oper. Theory

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On the Spectrum of the Schrdinger Hamiltonian of the One-Dimensional Harmonic Oscillator Perturbed by Two Identical Attractive Point Interactions

Cited by 35 publications

References 20 publications

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin II

Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive δ-impurities symmetrically situated around the origin

The Spectrum of a Harmonic Oscillator Operator Perturbed by $${\delta}$$ δ -Interactions

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