Displaced Harmonic Oscillator V ∼ min [(x + d)2, (x − d)2] as a Benchmark Double-Well Quantum Model
Round 1
Reviewer 1 Report
This is an interesting article. The author carefully studied bound-state quantum mechanics with oscillator-like potentials.
However, the paper does not say anything about physical systems to which his mathematical analysis is applicable. Thus, the author should mention this aspect, even for decorative purposes.
Author Response
> the paper does not say anything about physical systems to which his mathematical
> analysis is applicable.
> Thus, the author should mention this aspect, even for decorative purposes.
I appreciated and accepted the advice.
In revision I replaced, therefore, the rather too concise
first paragraph of Section 5 (Conclusions), viz., the text
``In our present paper we sought for a nontrivial support of the
existing tendencies towards an extension of the class
of ``solvable'' 1D potentials $V(x)$ from completely analytic to
non-analytic at some points and, in particular, in the origin. We
believe that these tendencies are natural and well motivated.``
by its following amended version (accompanied also by the
three new ``physics-oriented'' references added):
``In conclusion, let us re-emphasize that in the current literature
there exists a large number of papers and studies
devoted to the experiments and
realistic models of quantum physical systems
described by a simplified, one-dimensional (i.e., ordinary differential)
Schr\"{o}dinger equation (i.e., often, by our Eq.~(\ref{SEx}) which is
defined on the whole real line of the coordinate).
Still, in contrast to the dominance
of the scene by the papers
describing the bound states lying in a
single-well potential, one finds
perceivably less papers dealing with the potentials having
a double-well shape.
Naturally, one of the reasons is purely formal: One often finds that
many methods of solving Schr\"{o}dinger equations
(and, among them, the perturbation-expansion methods)
happen to fail when applied to the double-well systems.
This leads to a remarkable conflict between the prevailing, purely
mathematically motivated preference of the single-well models in
various methodical considerations, and the frequently encountered lack of
some sufficiently efficient construction recipes,
especially in the context of the double-well
scenarios which are much more interesting,
in the phenomenological context, mainly due to the emerging
manifestations of the
influence of the tunneling effect.
In such a context we sought,
in our present paper, for a nontrivial support of the
existing tendencies towards an extension of the class
of the ``exactly solvable'' 1D potentials $V(x)$
from their completely analytic subset to its
non-analytic generalizations.
We
are persuaded that such a tendency is natural and well motivated.
We also believe that in the nearest future our approach might find
some immediate and
successful applications, e.g.,
during some of the popular and, usually, partially non-numerical
descriptions of the physical systems
(as studied, e.g., in \cite{avocro}) in which,
in the language of mathematics,
the double-well structure of the potential
$V(x)$ is necessary, but in which
the phenomenon of
the avoided level crossing is expected to occur while being
not so easily detected
by the mere finite-precision
experimental measurements of the spectra.
Last but not least let us mention that among the physical systems to which
our present methodical innovation could also appear,
in the nearest future, to be immediately
applicable are also, potentially, all of the various
more-dimensional realistic
multi-well models as encountered, e.g.,
in quantum chemistry \cite{reloca}
and/or in the quantum-computation-motivated
mesoscopic physics of
the systems of quantum dots \cite{aloca}.
Indeed, under the widespread interaction-analyticity constraint,
just the approximate solutions of the
partial differential Schr\"{o}dinger equations are
usually available in these systems.''
\bibitem{avocro}
Miloslav Znojil, Avoided level crossings in quasi-exact approach. Nucl. Phys. B 967 (2021) 115431 DOI:10.1016/j.nuclphysb.2021.115431
(arXiv:2104.12144)
\bibitem{reloca}
Miloslav Znojil, Relocalization switch in a triple quantum dot
molecule in 2D . Modern Physics Letters B, Volume: 34 Issue: 33 Article Number:
2050378 Published: NOV 30 2020 (arXiv:2005.06934)
\bibitem{aloca}
Polynomial potentials and coupled quantum dots in two and
three dimensions By: Znojil, Miloslav ANNALS OF PHYSICS Volume: 416
Article Number: 168161 Published: MAY 2020
Reviewer 2 Report
M. Znojil studies the double displaced harmonic oscillator. This is a useful potential function to study with applications in atomic and molecular physics and solid state physics. The author demonstrates that even with the non-analyticity in this function, we can obtain non-polynomially exact solutions (NES) in specific regimes and can construct quasi-exactly solvable (QES) solutions for the rest. The paper is well written and contributes to the solution space of a model potential with use cases. I would recommend publication as is.
Author Response
Thanks for recommendation.
