Time-evolution of quantum systems via a complex nonlinear Riccati equation. II. Dissipative systems

Cruz, Hans; Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Óscar

doi:10.1016/j.aop.2016.07.029

Cited by 27 publications

(31 citation statements)

References 44 publications

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“…These last properties facilitate the interpretation of V I (x) as a rightful perturbation (for |λ| << 1) which would be associated with dissipation (see e.g. [39,40]).…”

Section: Pt -Symmetric Potentialsmentioning

confidence: 88%

Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum

Jaimes-Nájera

Rosas-Ortiz

2017

Annals of Physics

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Some general properties of the wave functions of complex-valued potentials with real spectrum are studied. The main results are presented in a series of lemmas, corollaries and theorems that are satisfied by the zeros of the real and imaginary parts of the wave functions on the real line. In particular, it is shown that such zeros interlace so that the corresponding probability densities ρ(x) are never null. We find that the profile of the imaginary part V I (x) of a given complex-valued potential determines the number and distribution of the maxima and minima of the related probability densities. Our conjecture is that V I (x) must be continuous in R, and that its integral over all the real line must be equal to zero in order to get control on the distribution of the maxima and minima of ρ(x). The applicability of these results is shown by solving the eigenvalue equation of different complex potentials, these last being either PT -symmetric or not invariant under the PT -transformation.

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“…These last properties facilitate the interpretation of V I (x) as a rightful perturbation (for |λ| << 1) which would be associated with dissipation (see e.g. [39,40]).…”

Section: Pt -Symmetric Potentialsmentioning

confidence: 88%

Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum

Jaimes-Nájera

Rosas-Ortiz

2017

Annals of Physics

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“…As a consequence, at the quantum level this model has generated quite a dispute on whether it can describe a dissipative system without violating the Heisenberg uncertainty principle; we refer to e.g. the discussion in [29][30][31][32][33][34] and references therein.…”

Section: Time-dependent Hamiltonian Systemsmentioning

confidence: 99%

“…We remark that η as in (34) is the standard (natural) contactification of a symplectic manifold whose symplectic structure is exact, as defined e.g. in [28] and that the second expression in (34) directly implies that in these coordi-…”

Section: Time-independent Contact Hamiltonian Mechanicsmentioning

confidence: 99%

“…In the preceding sections we have introduced the contact phase space for time-independent mechanical systems, equipped with the local coordinates (q a , p a , S), called contact coordinates. In these variables the equations of motion are expressed in terms of the contact Hamiltonian equations (37)-(39) and the contact form is expressed as in (34). As in the symplectic case, we are now interested in introducing those transformations that leave the contact structure unchanged, which are known as contact transformations [14,39].…”

Section: Contact Transformations and Liouville's Theoremmentioning

confidence: 99%

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Contact Hamiltonian mechanics

2017

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In this work we introduce contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and show that it is a natural candidate for a geometric description of non-dissipative and dissipative systems. For this purpose we review in detail the major features of standard symplectic Hamiltonian dynamics and show that all of them can be generalized to the contact case. (Alessandro Bravetti), hans@ciencias.unam.mx (Hans Cruz), diego.tapias@nucleares.unam.mx (Diego Tapias) 4 Conclusions and perspectives 29 Appendix A Invariants for the damped parametric oscillator 32 Appendix B Equivalence between the contact Hamilton-Jacobi equation and the contact Hamiltonian equations 34

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“…and x 2 =η(t) the (dimensionless) classical momentum [167][168][169][170]. The time-dependent coefficient of the quadratic term obeys the Riccati equatioṅ…”

Section: Time-dependent Oscillator Wave Packetsmentioning

confidence: 99%

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

Rosas-Ortiz

2019

Integrability, Supersymmetry and Coherent States

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A short review of the main properties of coherent and squeezed states is given in introductory form. The efforts are addressed to clarify concepts and notions, including some passages of the history of science, with the aim of facilitating the subject for nonspecialists. In this sense, the present work is intended to be complementary to other papers of the same nature and subject in current circulation.

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Time-evolution of quantum systems via a complex nonlinear Riccati equation. II. Dissipative systems

Cited by 27 publications

References 44 publications

Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum

Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum

Contact Hamiltonian mechanics

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

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