Quantum square-well with logarithmic central spike

Abstract

Singular repulsive barrier V (x) = −gln(|x|) inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction ℒeff(x) = −gln[ψ∗(x)ψ(x)] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small g or after an amendment of the unperturbed Hamiltonian. At any spike strength g, the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables x = expy which interchanges the roles of the asymptotic and central boundary conditions.

Full text

arXiv:1712.03672v1 [quant-ph] 11 Dec 2017
Quantum square well with logarithmic central spike
Miloslav Znojil1and Iveta Semor´adov´a2
Nuclear Physics Institute of the CAS, Hlavn´ı 130, 250 68 ˇ
Reˇz, Czech Republic
Keywords:
.
state-dependence of interactions;
effective Hamiltonians;
logarithmic nonlinearities;
linearized quantum toy model;
PACS number:
.
PACS 03.65.Ge Solutions of wave equations: bound states
Abstract
Singular repulsive barrier V(x) = −gln(|x|) inside a square well is interpreted and studied as
a linear analogue of the state-dependent interaction Leff (x) = −gln[ψ∗(x)ψ(x)] in nonlinear
Schr¨odinger equation. In the linearized case, Rayleigh-Schr¨odinger perturbation theory is shown
to provide a closed-form spectrum at the sufficiently small gor after an amendment of the un-
perturbed Hamiltonian. At any spike-strength g, the model remains solvable numerically, by the
matching of wave functions. Analytically, the singularity is shown regularized via the change
of variables x= exp ywhich interchanges the roles of the asymptotic and central boundary
conditions.
1znojil@ujf.cas.cz
2semoradova@ujf.cas.cz
1

1 Motivation
The study of complicated quantum systems may be facilitated by a judicious, less explicit treat-
ment of certain less essential degrees of freedom [1]. The reduction may lead to simplifica-
tions which are often achieved via a tentative replacement of the exact, full-Hilbert-space linear
Schr¨odinger equation
i∂t|Ψi=H|Ψi(1)
by its reduced, open-subsystem version. In this manner a remarkably successful description of
the physical reality has been achieved, in multiple phenomenological applications (cf., e.g., their
compact review in paper I [2]) via nonlinear evolution equations in which the effective-interaction
potential was admitted state-dependent,
i∂tψ(x, t) = −∂2
x+V(eff)(x, t)ψ(x, t), V(eff)(x, t) = V[ψ(x, t), x, t] (2)
and, in particular, in which the nonlinearity has been chosen logarithmic,
V[ψ(x, t), x, t] = −gln[ψ∗(x, t)ψ(x, t)] , g > 0.(3)
In paper I it has been argued that an important formal support of the latter choice may be seen in
its asymptotic system-confinement self-consistency. Indeed, one can easily verify that the insertion
of a tentative, exactly solvable harmonic-oscillator potential V(HO )(x) = x2in Eq. (2) would yield
the wave functions in closed form
ψ(HO)(x, t)∼exp(−iE(HO)t) exp −x2/2 + O(ln |x|),|x| ≫ 1.(4)
In turn, expression (3) will lead to the qualitatively correct asymptotics
−ln[ψ∗
(HO)(x, t)ψ(HO)(x, t)] = x2+O(ln |x|),|x| ≫ 1 (5)
of the confining potential.
In the constructive part of paper I it has been recalled and demonstrated that certain node-less,
“gausson” solutions of the evolution Eq. (2) may be well-behaved at all times t. The existence of
the “gaussons” can be viewed as a consequence of the absence of the nodal zeros in the initial (i.e.,
say, t= 0) choice of the ground-state-like wave function ψ(x, 0). In other words, what remained
unclarified in paper I was the question of the properties of all of the non-gausson solutions of
Eq. (2) + (3). Presumably, most of these non-gausson solutions might be based on the anomalous
initial wave functions ψN(x, 0) having an N−plet of the excited-state-like nodal zeros at some
N≥1.
2

Naturally, this would make the nonlinear interaction term (3) singular. Locally (i.e., out of the
asymptotic region and near any nodal zero xjwith 1 ≤j≤N) this follows from the simple-zero
estimate
ψN(x, 0) ∼x−xj.(6)
In the first nontrivial non-gausson case we may choose N= 1 and require x1= 0 (i.e., an
initial-time antisymmetry of the wave function). In the light of the above-mentioned HO example
one expects that, with certain implicit, not too well tractable error terms, we should work with
potentials of the form
V(eff)(x, 0) ∼x2−2gln |x|+... (7)
or, in an alternative, technically simpler confining square-well (SW) approximation, with
V(eff)(x, 0) ∼V(SW )(x) = (∞,|x| ≥ 1,
−2gln |x|,|x|<1.(8)
In what follows we intend to complement the global non-linear-theory considerations of paper I
by the missing and relevant discussion of some of the technical consequences of the emergence of
the logarithmic singularity at one or more nodal points xjand, first of all, of certain properties of
the bound states in the first nontrivial excitation-simulating linear interaction potential (8).
2 Weak-coupling regime
The enormous simplicity of the one-dimensional quantum square well oscillator makes it suitable
for pedagogical purposes. Its analyses appear not only in conventional textbooks [3, 4] but also in
the less conventional studies of supersymmetric quantum systems [5, 6]. The elementary nature of
the square-well model found also nontrivial applications in parity times time-reversal symmetric
quantum mechanics [7, 8, 9, 10] or in certain sophisticated versions of perturbation theory [11].
We intend to analyze the interaction in its special form (2) + (8), i.e., in the first nontrivial
N= 1 special case. Thus, our attention gets shifted to the linearized model represented by
the conventional and time-independent ordinary differential Schr¨odinger equation for quantum
stationary bound states,
−d2
dx2ψn(x)−2gln(|x|)ψn(x) = Enψn(x), ψn(±1) = 0 , n = 0,1,... . (9)
The spike-shaped logarithmic barrier is unbounded, repulsive and centrally symmetric here - cf.
Fig. 1 where the shape of the potential is displayed at g= 0.25,0.5 and 1.
It is worth adding that although we are choosing here (i.e., in Eq. (6)) the first nontrivial nodal
number N= 1 for the sake of simplicity, Eq. (9) describes all of the bound states generated by the
3

0
5
10
–1 0 1
1
0
x
V(x)
Eψ
ψ
Figure 1: The shapes of the perturbed square-well potentials (8) at g= 1/4 (the lowest, light
spike), g= 1/2 (the intermediate spike) and g= 1 (the upper spike). Horizontal lines mark the
first two energy levels for g= 0. Attached to them, the picture also displays the shapes of the
unperturbed wave functions.
linearized toy-model interaction (8). These states are numbered by a different, lower-case index
n. Naturally, one would have to set, for the sake of consistency, n=Nat the end of the analysis
and, in principle at least, before an ultimate return to the initial nonlinear-equation setting.
2.1 First-order approximation
At the small but non-vanishing strengths g > 0 our central barrier is infinitely high so that one
might suspect that it is impenetrable. The expectation is wrong. In the Rayleigh-Schr¨odinger
perturbation-expansion series arrangement [12] with
En=En(g) = E[0]
n+g E[1]
n+g2E[2]
n+. . . , E[0]
n= [(n+ 1)π/2]2, n = 0,1,... (10)
the first-order shifts
E[1]
2p∼ −4Z1
0
cos2[(2p+ 1)π x/2] ln(|x|)dx , p = 0,1,... (11)
and
E[1]
2q+1 ∼ −4Z1
0
sin2[(q+ 1)π x] ln(|x|)dx , q = 0,1,... (12)
of the spectrum of our conventional perturbed square well are, obviously, positive and finite.
One can easily prove that the latter corrections are all finite, indeed. The proof may be based
on the fact that all of the integrands are bounded on the subintervals of x∈(ǫ, 1) with any
ǫ= exp(−R)∈(0,1) (i.e., any R > 0). Thus, it is sufficient to find a bound for the integrals over
4

a short interval of x∈(0, ǫ). With a sufficiently large R≫1 we obtain an explicit estimate
−4Zǫ
0
sin2[(q+ 1)π x] ln(|x|)dx ∼Zǫ
0
x2ln(|x|)dx =Z−R
−∞
y e3ydy =−1
9(3R+ 1) e−3R(13)
showing that the odd-state integrals are all exponentially small. In the even-state case we have,
similarly, the estimate
−4Zǫ
0
cos2[(2p+ 1)π x/2] ln(|x|)dx ∼Zǫ
0
ln(|x|)dx =Z−R
−∞
y eydy =−(R+ 1) e−R(14)
leading to the same ultimate finite-correction conclusion.
2.2 Closed formulae
The even-state contribution seems larger than the odd-state contribution, for the sufficiently
large Rat least. For a more reliable, R−independent comparison of the corrections E[1]
nat
the even and odd nit is necessary to introduce the conventional sine-integral special functions
Si(x) = Rx
0sin(t)/t dt and to evaluate the first-order corrections exactly. Fortunately, this is
feasible yielding the following formulae,
E[1]
2p= 2 + 2
(2p+ 1)πSi[(2p+ 1)π], p = 0,1,... , (15)
E[1]
2q+1 = 2 −2
(2q+ 2)πSi[(2q+ 2)π], q = 0,1,... . (16)
The evaluation of the numerical values of the special functions Si(x) is routine and yields
the first-order perturbation-series coefficients as sampled in Table 1. The inspection of the Table
reveals that the energy shifts at the even quantum numbers n= 2pwill be always larger than the
partner shifts at the odd quantum numbers n= 2p+ 1. In other words, even the present “soft”,
logarithmic shape of the central repulsive barrier will lead to the quasi-degeneracy pattern in the
spectrum of the logarithmically spiked bound states.
2.3 Limitations of applicability
By the logarithmically singular but still positive-definite barrier the unperturbed spectrum is
being pushed upwards. Still, due to the immanent weakness of the singularity of the logarithmic
type even in the strong-coupling dynamical regime with g≫1, the expected effect of the quasi-
degeneracy will get quickly suppressed with the growth of n, i.e., of the excitation. In contrast to
the stronger and more common (e.g., power-law) models of the repulsion in the origin, this will
make the real influence of the logarithmic barrier restricted to the low-lying spectrum.
5

Table 1: The first ten numerical coefficients E[1]
nin Eq. (10).
nsymmetric ψn(x) antisymmetric ψn(x)
0 3.178979744
1 1.548588333
2 2.355395491
3 1.762515165
4 2.208042866
5 1.838931594
6 2.146975999
7 1.878156443
8 2.113606700
9 1.902022366
Fig. 2 may be recalled for an explicit quantitative illustration of the latter expectation. Using
just the most elementary leading-order-approximation estimates we see there that while the pre-
diction of the quasi-degeneracy between the ground (i.e., n= 0) and the first excited (i.e., n= 1)
state might still occur near the reasonably small value of coupling g[1]
0,1= 4.540138798, the next
analogous crossing of the first-order energies E[1]
2and E[1]
3only takes place near the estimate as
large as g[1]
2,3= 29.13203044, etc.
0
20
40
60
80
10 20 30
g
E
Figure 2: The coupling-dependence of the low-lying spectrum in the Rayleigh-Schr¨odinger first
order (i.e., linear-extrapolation) approximation.
The same Fig. 2 also shows that another first-order crossing may be detected for E[1]
0and E[1]
2,
emerging even earlier (i.e., at g[1]
0,2= 23.96744320) and being, obviously, spurious. In other words,
for the prediction of the quasidegeneracy in the strong-coupling dynamical regime the knowledge
6

of the mere first-order perturbation corrections must be declared insufficient.
3 Strong-coupling regime
Beyond the domain of applicability of the weak-coupling perturbation theory, alternative (mainly,
purely numerical) methods must be used in order to determine the spectrum of bound states of
our linearized toy model exactly, i.e., with arbitrary prescribed precision.
As long as these bound states are determined by the ordinary differential Schr¨odinger Eq. (9),
there exists a number of methods of their construction. The choice of the method may be inspired
by the inspection of Figs. 1 and 2. This indicates that the influence of our singular logarithmic
potential (8) is felt, first of all, by the low-lying bound states and/or in the strong-coupling regime.
Directly, this may be demonstrated by the routine numerical construction of the wavefunctions
(sampled in Fig. 3) and by the routine numerical evaluation of the energies (sampled in Table 2).
In both cases, due attention must be paid to the singular nature of our potential (8) in the origin.
This is a challenging aspect of the numerical calculations which will be discussed and illustrated
by some examples in what follows.
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure 3: The first three wavefunctions at g= 10.
3.1 Mathematical challenge: singularity
Without any real loss of generality our attention may stay restricted to the characteristic N= 1
model (9). This model is sufficiently general when treated as a system of two (viz., x < 0
and x > 0) linear differential equations for which the logarithmic derivatives of the respective
7

Table 2: Numerically determined low-lying spectrum.
g E0E1E2E3E4
0.00 2.4674 9.8696 22.207 39.478 61.685
0.25 3.2478 10.255 22.796 39.928 62.265
0.50 4.0097 10.638 23.394 40.369 62.817
1.00 5.4784 11.395 24.618 41.252 63.931
wave functions have to be matched in the origin, i.e., at x1= 0. In this sense, naturally, the
generalization to the N > 1 cases would be straightforward.
Once we fix N= 1 we may split the original linear differential equation into a pair living on
the two respective half-intervals, viz.,
−d2
dx2
(L)
ψn(x(L))−2gln(−x(L))ψn(x(L)) = Enψn(x(L)), x(L)∈(−1,0) (17)
and
−d2
dx2
(R)
ψn(x(R))−2gln(x(R))ψn(x(R)) = Enψn(x(R)), x(R)∈(0,1) .(18)
As long as the bound-state solutions have a definite parity at N= 1, one of these equations may
be omitted as redundant.
3.2 Amended zero-order approximation
Due to the symmetry of our potential it will be sufficient to consider just Eq. (17) on the negative
finite half-interval. The correct matching with Eq. (18) may be then guaranteed by means of the
pair of ad hoc boundary conditions in the origin,
(ψn(0) 6= 0 , ψ′
n(0) = 0 ,(even states) ,
ψn(0) = 0 , ψ′
n(0) 6= 0 ,(odd states) .(19)
The application of recipe (19) must be performed with due care. The reason lies in the unbounded,
singular nature of our logarithmic potential in the origin. Fortunately, the necessary deeper
analysis of the matching conditions at x= 0 might be also complemented by an amendment of
the perturbation-theory considerations of section 2.
Inside a suitable small interval of x∈(−d, d) with d < 1 we may contemplate an approximative
replacement of the logarithmic repulsive spike V(x) by a rectangular barrier of a finite height
κ2
0=E+κ2. At any given/tentative energy Ewe may demand that E=k2=−2gln d, i.e., that
8

d=d(E) = exp[−E/(2g)]. This will weaken our potential V()x) to the left from x=−dwhile
strengthening it, locally at least, to the right from x=−d. The approximate wavefunctions may
be then written down in the following closed elementary-function form
ψ(x)≈








sin[(x+ 1)k], x ∈(−1,−d),
sin[(x−1)k], x ∈(d, 1) ,
cosh[κx], x ∈(−d, d) (even states) ,
sinh[κx], x ∈(−d, d) (odd states) .
(20)
Its components must be properly matched at x=±d=±d(E) and, if needed, also properly
normalized. It is, perhaps, worth adding that inside the interval of E∈(5,5.5) (i.e., in the vicinity
of the gound-state energy, cf. Table 2), the decrease of the value of d(E) from d(5) ≈0.08208499862
to d(5.5) ≈0.06392786121 is almost linear, i.e., comparatively easily kept under control in the
calculations. This means that the d−dependence of our approximative rectangular barrier may
be also re-interpreted as its energy-dependence.
3.3 Regularized Schr¨odinger equation
The remarkable technical challenge is that the point of the matching of the wavefunctions coincides
with the singularity of the potential. In the literature, similar situation is usually encountered in
connection with the Coulombic repulsion ∼1/x [13, 14]. Here, the repulsion is much weaker but
the regularization of the model is still by far not routine. In the analytic-function-theory language
it can rely upon the change of variables
x(L)=−exp(−λ), x(R)= exp(ρ),(21)
and
ψn(x(L)) = exp(−λ/2) φ(−)(λ), ψn(x(R)) = exp(ρ/2) φ(+)(ρ).(22)
In the left subinterval of x(L)∈(−1,0) this yields the initial-value problem
−d2
dλ2φ(λ) + 1
4φ(λ) = exp(−2λ) [E−2gλ]φ(λ), φ(0) = 0 , φ′(0) = 1 (23)
on the half-line of λ∈(0,∞).
We see that the singularity has been moved to infinity. Thus, after the change of variables (22),
the wavefunction-matching relations (19) may purely formally be replaced by their appropriate
asymptotic analogues supplementing Eq. (23). In fact, we shall show below that this is really a
purely analytic idea, not leading to any practical numerical advantages.
9

Analogously, the second half of our initial differential equation with x(R)∈(0,1) is converted
into equivalent second half-line version
−d2
dρ2φ(ρ) + 1
4φ(ρ) = exp(2ρ) [E+ 2gρ]φ(ρ), φ(0) = 0 , φ′(0) = (−1)n+1 =±1 (24)
where ρ∈(−∞,0). The latter equation is redundant again, obtainable from the former one by
the mere change of parity alias replacement ρ→ −λ.
4 Numerical solutions
Using our amended approximation (20) we could obtain a qualitatively correct shape of the wave-
functions, in principle at least. Naturally, the fully reliable construction of bound states must be
performed by the controlled-precision numerical integration of our ordinary differential Schr¨odinger
equations. These results were sampled in Fig. 3 above. What is worth emphasizing is that in this
setting the singularity of potential V(x) remains tractable by the standard numerical-integration
software, say, of MATHEMATICA or MAPLE.
4.1 Qualitative theory: re-parametrized Schr¨odinger equation
For our model (9), obviously, an extension of applicability of the conventional perturbation theory
could have been based on the use of various more sophisticated zero-order shapes of V0(x). In
particular, we could obtain a more quickly convergent sequence of perturbation approximations
when using the specific rectangular-potential choice of V(RP )
0(x) of paragraph 3.2.
One of the other benefits of the amendment would be qualitative, based on the observation
that a small deformation of the special potential V(RP )
0(x) of paragraph 3.2 must lead just to a
small deformation of the related trigonometric definitions (20) of the wavefunctions.
Via these deformations, we may even return exactly to our full-fledged logarithmic potential
V(x). Such a return would be achieved by means of the replacement of the effective-momentum
constant k=√Eby a weakly coordinate-dependent function µ(E, x) = pE−V(x). The other
constant κ=κ(E) gets replaced by the effective barrier-height ν(E, x) = pV(x)−E. As a
result, this enables us to rewrite our Schr¨odinger equation in the partitioned form
−d2
dx2ψ(x) = (µ2(E, x)ψ(x), x ∈(−1,−d)S(d, 1) ,
−ν2(E, x)ψ(x), x ∈(−d, d).(25)
By construction, this equation is exact. Still, in the light of Eq. (20) it may be assigned the
10

approximate elementary solutions
ψ(x)≈ ± 








sin[µ(E, x)(x+ 1)] , x ∈(−1,−d),
sin[µ(E, x)(x−1)] , x ∈(d, 1) ,
cosh[ν(E, x)x], x ∈(−d, d) (even states) ,
sinh[ν(E, x)x], x ∈(−d, d) (odd states) .
(26)
Once we take into account the parity, the matching in the origin remains trivial. The decisive role
will now be played, instead, by the smoothness (i.e., sort of matching) of the amended approximate
wave functions (26) at the energy-dependent point x=−d < 0 or, equivalently, at x=d > 0.
4.2 The danger of ill-conditioning
The wavefunction ψ0(x) = ψ(x) is nodeless and spatially symmetric. It must obey the ordinary
differential Eq. (17) on half-interval so that its construction may proceed numerically. This yields
the x→0−limiting values which must be made compatible with the upper line of Eq. (19) via a
suitable choice of energy E. Such an algorithm the leads to the results sampled by Table 2.
In principle, we could also employ the change of variables (21) and (22) and replace Eq. (17)
by its equivalent version (23). This enables us to transfer the central left-right matching (19) of
wavefunctions to its analogue in infinity, i.e., in the limit λ→+∞.
The key merit of such an arrangement lies in the clear picture of the analyticity properties and,
in particular, in the asymptotic negligibility of the right-hand side of Eq. (23). For this reason
the new equation may be immediately assigned the elementary general asymptotic solution
φ(λ, E) = const eλ/2+corrections +f(E)e−λ/2+corrections . λ ≫1.(27)
The changes of the energy only influence here the subdominant term which is exponentially small.
This means that any numerical solution of the initial-value problem (23) will remain insensitive
to the variations of the tentative energy. Hence, the task is ill-conditioned.
One has to try to start the reconstruction in the opposite direction initiated at a sufficiently
large λmax ≫1 via a suitable tentative choice of the initial values of the wavefunction φ(λ) and
of its first derivative.
In a test study of the ground state at g= 1 we made use of our knowledge of the first-
order perturbation result E≈(π/2)2+ 3.178979744 = 5.646380845 of section 2. Unfortunately,
any choice of the asymptotic initial values of wavefunctions seems to be destabilized by certain
uncontrolled numerical rounding errors as well. This is an empirical observation sampled in Table 3
in which we employed the simplest possible choice of the tentative asymptotic initialization
φ(λmax) = 1 , φ′(λmax) = 0 .(28)
11

Table 3: The failure of the backward-iteration method based on Eqs. (23) and (28). At q= 1, the
convergence of φ(0) →0 with respect to λmax → ∞ is too slow.
E λmax evaluated φ(0) difference
5.55 3.50 -0.064333935 -
3.75 -0.059104634 -0.0052
4.00 -0.053830824 -0.0053
4.25 -0.048788380 -0.0050
5.45 3.50 -0.037417250 -
3.75 -0.031754144 -0.0057
4.00 -0.026113710 -0.0056
4.25 -0.020763014 -0.0053
The inspection of the Table reconfirms the scepticism evoked by the analytic formula for wave-
function asymptotics (27) which are numerically ill-conditioned.
5 Discussion
In the nonlinear Schr¨odinger equation context as formulated and reviewed in paper I the restriction
of the constructive attention to the mere node-less gaussons (i.e., to N= 0) really weakened the
authors’ original intention of making the asymptotically confining interaction (3) truly excitation-
dependent (i.e., more precisely, number-of-nodal-zeros-dependent).
On the basis of results of the preceding section we may now conjecture that in practice the
true impact of the presence of the nodal zeros will be probably much smaller than expected.
Although these zeros induce the infinitely high barriers in the nonlinear effective potentials (3),
these barriers remain penetrable and narrow.
We saw here that in the context of perturbation theory the latter properties of the logarithmic
barriers render the quantitative considerations feasible. The surprising, not entirely expected
friendliness of the perturbation analysis of toy model (9) encourages also the use of the other,
more universal numerical methods.
5.1 Linear models with N > 1
One of our key results may be seen in the observation that in the technical sense one need not
feel afraid of the presence of the logarithmic (i.e., as we demonstrated, weak) singularities in the
12

interaction potentials, linear or not. In particular, in the linear case one may feel encouraged to
employ the standard techniques of the matching of the piecewise analytic wave functions at the
nodal points xj. In this context the readers may be recommended to have a look at a few recent
constructive analyses [15, 16] of the similar scenarios.
In our present note we skipped the concrete numerical implementation of the matching recipe.
We have only pointed out that due to the logarithmic nature of the singularity of the potential
(8), one has to keep in mind that the most natural change of variables (21) + (22) transforms the
origin of x(L,R)into infinities of λand ρ, and vice versa. Still, we believe that this would cause
just a minor complication in numerical setting, more than compensated by the simplification of
the differential equation.
After the change of the variables, all of the basic features of the conventional matching method
remain unchanged. As long as in the new setting of Eqs. (23) and/or (24) the energy-representing
parameter Ebecomes multiplied by an exponential function, one should speak, strictly speaking,
about the so called Sturmian eigenvalue problem.
5.2 Towards the nonlinearities
In the majority of the phenomenological scenarios, the predictions provided by the guess of the
linear interaction V(~x) may fail to fit the reality sufficiently closely. One of the main reasons
is that in practice (i.e., up to a few most elementary quantum systems), the physics behind the
interaction often proves complicated: relativistic or nonlocal or nonlinear.
In our present text we repeatedly pointed out that our present linearized and perturbed square-
well model is certainly interesting per se. It might fulfill the role of an interesting effective model
in physics. Still, its methodical relevance is related to the nonlinear setting of paper I, poten-
tially useful in the context of the study of quantum systems described by certain prohibitively
complicated conventional linear Schr¨odinger equations (1).
One of the fairly instructive testing grounds of the efficiency of the suppression of the tech-
nical complications via non-linearization may be found in classical optics where certain deeply
relevant dynamical effects may be very efficiently described via a transition V(~x)→V(ψ(~x)) to a
suitable state-dependent interaction term. With one of the least complicated tentative choices of
V(ψ(~x)) ∼ψ∗(~x)ψ(~x) one arrives at the highly popular toy model called “non-linear Schr¨odinger
equation” [17, 18, 19]. In this context, we tried to find a formal encouragement and support also
for the logarithmic self-interaction in our present letter.
In practice, typically, the strictly linear theory only remains friendly and feasible for the most
elementary systems like hydrogen atoms, etc. Moreover, even in the phenomenology based on the
linear equations one of the key roles is played by the educated guess or knowledge of the relevant
13

dynamical input information about the linear interaction. Thus, one may conclude that in this
language the transition to the effective nonlinear models (including (3)) does not look drastic or
counterintuitive to a physicist.
A number of supportive phenomenological arguments may be found in paper I or, e.g., in the
recent remark [20] on the effective nonlinear logarithmic Schr¨odinger equations
i∂tψ(~x, t) = (−∆ + VLSE )ψ(~x, t), VLSE =−bln |ψ(~x, t)|2,(29)
where their relevance in the phenomenology of quantum liquids has been emphasized. Alas, the
situation may become perceivably more complicated in mathematics. Multiple challenges emerge
there. In their light, our present letter may be perceived as constructive commentary on these
complications, i.e., as a contribution to a future completion of the formalism of practical quantum
mechanics [3].
Acknowledgements
The project was supported by GA ˇ
CR Grant Nr. 16-22945S. Iveta Semor´adov´a was also supported
by the CTU grant Nr. SGS16/239/OHK4/3T/14.
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