The quantum Arnold transformation and the Ermakov–Pinney equation

Guerrero, Julio Becerra; López-Ruiz, Francisco F.

doi:10.1088/0031-8949/87/03/038105

Cited by 21 publications

(33 citation statements)

References 30 publications

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“…Finally, the approach can be extended either by applying conventional 1‐step Darboux transformations on the complex‐valued potential V λ ( x ) or by iterating the procedure presented in this work. Further insights may be achieved from the Arnold and point transformations . Results in these directions will be reported elsewhere.…”

Section: Discussionmentioning

confidence: 99%

“…Further insights may be achieved from the Arnold and point transformations. [26][27][28][29] Results in these directions will be reported elsewhere.…”

Section: Discussionmentioning

confidence: 99%

“…The new potential is shown in Figure 3 for = 0 and 2 different values of N. Notice that ReV (x) is very close to the Morse potential at the edges of R. Besides, such a function exhibits a very localized deformation that serves to host the additional energy E 0 = 0. In turn, ImV (x) satisfies the condition of zero total area (26). Clearly, these potentials are not eligible for PT transformations.…”

Section: Morse Potentialsmentioning

confidence: 99%

“…The former case is shown in Figure 7 for 2 different values of the parameter r. In both cases, ReV (x) is even, with a very localized symmetric deformation that permits the presence of the new energy (in the figure, = 1∕4). In turn, ImV (x) is odd and satisfies the condition of zero total area (26). That is, the spatial reflection of ImV (x) is compensated by the complex conjugation (i → −i).…”

Section: Figurementioning

confidence: 99%

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Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Blanco-Garcia

Rosas-Ortiz

Zelaya

2018

Math Methods in App Sciences

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Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.

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Section: Discussionmentioning

confidence: 99%

“…Further insights may be achieved from the Arnold and point transformations. [26][27][28][29] Results in these directions will be reported elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Section: Morse Potentialsmentioning

confidence: 99%

Section: Figurementioning

confidence: 99%

See 2 more Smart Citations

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Blanco-Garcia

Rosas-Ortiz

Zelaya

2018

Math Methods in App Sciences

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“…At the classical level this so-called Arnold transformation [31] was introduced in order to transform a generic second-order differential equation, that physically describes a driven harmonic oscillator with time-dependent friction coefficient and time-dependent frequency, into the differential equation corresponding to the motion of a free particle. Its implementation as a unitary map at the quantum level has been investigated for instance in [15]. From our approach we can make an immediate connection to this formalism by going back to equation (2.14) that has played an important role in deriving our classical transformation.…”

Section: A Proposal Of a Modified Map For The Infrared Modes: The Armentioning

confidence: 99%

Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

2019

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We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quaside Sitter spacetime. Our main interest lies in the question to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation and we show that its generalization to Fock space has to be chosen appropriately in order that the Shale-Stinespring condition is not violated, where we also compare our results to already existing ones in the literature. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution to the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.

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Hermite Coherent States for Quadratic Refractive Index Optical Media

Gress

Cruz

2019

Integrability, Supersymmetry and Coherent States

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Ladder and shift operators are determined for the set of Hermite-Gaussian modes associated with an optical medium with quadratic refractive index profile. These operators allow to establish irreducible representations of the su(1, 1) and su(2) algebras. Glauber coherent states, as well as su(1, 1) and su(2) generalized coherent states, were constructed as solutions of differential equations admitting separation of variables. The dynamics of these coherent states along the optical axis is also evaluated.

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The quantum Arnold transformation and the Ermakov–Pinney equation

Cited by 21 publications

References 30 publications

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

Hermite Coherent States for Quadratic Refractive Index Optical Media

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