After S. Smale's works it became clear that in smooth dynamics the system of a general form is not structurally stable and therefore there is no strict mathematical basis for modeling and computational analysis of systems. The contradiction appeared in science: according to physicists dynamics is simple and universal. The solution to this problem was proposed based on the construction of dynamic quantum models (DQM). From the assumption that quantum effects are caused by unrecoverable “white noise”, a certain mathematical model of quantum mechanics already follows and is essentially unambiguous. On the other hand, in this model spectral problems are reduced tothe usual perturbation theory of smooth dynamical systems. Thus, the construction of such models can be considered as an asymptotic method for solving spectral problems. But the definition of DQM is not formally related to Hamiltonian systems. DQM is defined and constructed universally for both Hamiltonian systems and systems with the truth function. As a result, for example, quantization with the Bohr-Sommerfeld condition also extends to systems with a truth function. Hopefully DQM opens for new applications. The most important is to seek assistance and cooperation in future research.
Increasingly, processes and systems are researched or developed through computer simulations and this trend is likely to continue [1]. Computational modeling has been used in physics, chemistry and related engineering for many decades because this is the only way the equations can be solved at all [2]. It consists from two steps: (i) modeling, i.e. finding a model description of a real system, and (ii) solving the resulting model equations using computational methods [3].
But if an arbitrarily small perturbation of the model leads to a qualitatively different picture of the dynamics, then such a model is not applicable to the real process: strictly speaking, perturbations are included in the definition of a model [4]. Also computational methods inevitably lead to errors of discretization and rounding in calculations [5]. Therefore traditionally the stability of a model with respect to relaCopyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). tively small changes is a necessary condition for its correctness [6]. The qualitative invariance of a mathematical model under small perturbations is usually called structural stability [7].
However, in S. Smale's works [8] was shown, that there exist smooth dynamic systems whose neighborhoods do not contain any structurally stable system. This meant that there was no rigorous mathematical basis for modeling and computational analysis. The contradiction has appeared in science, because physicists believe that the dynamics is simple and universal [9].
The solution to this problem was proposed in [10] based on the construction of dynamic quantum models (DQM). It turned out that taking into account random fluctuations, necessary for the transition to the quantum model of reality, allows us to return in fact to the simple picture of A. Poincare’s dynamics: a dense set of structurally stable systems.
DQM is so named because for Hamiltonian systems it is simply related to the corresponding Schrödinger equation. From the assumption that quantum effects are caused by unrecoverable “white noise”, a certain mathematical model of quantum mechanics already follows and is essentially unambiguous [11]. Dynamics in it is described by Markov cascades (time is discrete). This model is simply connected with the traditional one: there is a simple correspondence between Markov cascades and quasisolutions of the corresponding Schrödinger equation. Thus, in a sense, DQM is a bridge between the formal calculus of quantum mechanics and the intuitive vision of physicists. On the other hand, in this model spectral problems are reduced to the usual perturbation theory of smooth dynamical systems. Thus, the construction of such models can be considered as an asymptotic method for solving spectral problems.
This paper gives an example of such approach to the one-dimensional system with the quasiperiodic potential (Proposition 2).
But the definition of DQM is not formally related to Hamiltonian systems; it is defined for any ordinary differential equation or any diffeomorphism on any smooth Riemannian manifold. Hopefully this opens the way for absolutely new applications.
For example, for applications to dynamic systems that using logical operations: algorithms, theorems, software applications. The use of fuzzy logic in DQM is in principle completely natural and even almost inevitable. Let J = J (z) be a given smooth function on phase space (0 J (z) 1), equal to 1 on the true trajectory and 0 outside some neighborhood of it; we can interpret it as the function of truth. In this paper, DQM is defined and constructed universally for both Hamiltonian systems and systems with the truth function J. As a result, for example, the point of the DQM spectrum is interpreted exactly as the average value of truth for approximate logical conclusions.
Quantization with the Bohr-Sommerfeld condition also extends to systems with a truth function (Proposition 1).
But the reverse is also true. If we construct an approximate model of given theorem or software application using the training of a neural network, then we get the DQM of these objects. With further training, these DQMs will approach the original object (in other words, they will converge to it according semiclassical limit). Perhaps this will allow a new approach to the problems of solvability in logic, there is some analogy with the theorem of the equivalence of structural stability and hyperbolicity proved in [10]. DQM of systems with logical operations are always uniformly limited by the number of operations (see Section 2.2) and then for them solvability is not in doubt.
Then everything depends on the semiclassical limit, more precisely, on the uniformity of the structural stability of DQM.
The paper goal is 1) to build the foundations of the theory of dynamic quantum models (DQM); 2) to demonstrate the application of this theory for spectral problems of quantum mechanics.
The paper is organized as follows: in part 2 we synthesize the dynamic quantum model (DQM); in part 3 we demonstrate the application of DQM for spectral problems of quantum mechanics; part 4 concludes.
We had to omit proofs of some propositions in order to fit the paper format. 2 2.1
Let p(x) be an n -dimensional smooth vector field on an n -dimensional smooth
Riemannian manifold M , where x(x1, x2 ,..., xn ) are local Euclidean coordinates on
M , pi (x) C (Rn ) ( i = 1,..., n ). On each phase curve x(t) M of the
dynamical system generated by this vector field
dxi = pi (х) , ( i = 1,..., n )
dt
(
So, the distance d traveled by a point along the path of (
Now suppose that the configuration space is one-dimensional, there is a turn point on the segment of the trajectory, the initial position is located near the turn point and move towards it. To pass a segment of the path, the ends of which are significantly different, the point must reach the turning point, and then (after turning) pass another segment of ρ – length 2 . How much will the distance to the nearest significantly different measurement increase? The points on the segment ρ – lengths 2 , including the turning point, are indistinguishable among themselves, only their average value is important. Therefore, the points in the interval between 3 2 and 1 2 to the turning 2 2 point in time 2 will move to the segment centered just on the turning point; and after a while until the next significantly different position. So, only for points on a segment with length 1 2 to a turning point, the distance to the nearest significantly 2 different measurement will increase: on average by 1 2 . 4
In the general case, one should take into account those points that move towards the caustic K – the set of singular points at which the direction of motion changes and are located at ρ – distance 1 2 from K. To pass a segment of the path, the ends 2 of which are significantly different, such points must reach K, and then (turning) to pass another segment of ρ – length 2 in a new direction. Therefore, the travel time for these points will increase on average by 1 2 . The jump in the time interval at 4 turning points on the optical time scale is a quantum-mechanical phenomenon traditionally taken into account by means of the Morse index (when establishing a connection with the Schrödinger equation, it turns out that 1 2 = h , i.e. 2 = 2πh). 4 2 Generally speaking, there may be features on the caustic other than turn points, however, such a singular point splits into several turn points with an arbitrarily small stir [6]. At such points, the number 1 2 = h is added to the ρ - length, where μ 4 2 is the Morse index of the singular point. Here μ is an integer equal to the number of turning points that arose during small stir and passed in the positive direction to the caustic minus the number of turning points traveled in the negative direction.
So, a dynamic quantum model first shifts each point along the phase curve of a
given dynamic system over the optical time 2 (or ρ – length 2 ) and in a
neighborhood of the caustic this shift increases abruptly by 1 2 = μ h . And then
4 2
randomly shifts on a distance not exceeding the length of the trajectory from the
original to the new point. The following rigorous definition summarizes this description.
The definition of a dynamic quantum model is given for an arbitrary dynamic system
(
Definition 1. By a dynamic quantum model (DQM) for dynamical system (
A where t is the shift time from x to Gx along the path of the ρ-length 2h or 2h + h in the neighborhood of the caustic, 2 = 2h . Given the initial distribu2 tion, we obtain a Markov process P with this initial distribution and the transition function P( y, A) : if t is the distribution at time t , t is the lag between the two nearest measurements, then the DQM sets new distribution P(t ) = t+t at time P( x, A) = 2t e−( y−Gx)2 / 2 2t dy , t + t . 2.2
Our goal is to determine pure states and eigenvalues of DQM. And now, along with the discreteness of the measurement process, its limited time will be essential. Of course, the measurement process cannot continue indefinitely, but here its duration is dictated by the very definition of DQM. Namely, the duration of the measurement, in principle, cannot exceed on order 1 since further the measurement errors with dish persion 2 t (where the diffusion coefficient 2 is small of order h) are no longer small and the notion of trajectory loses its meaning. (And you can only talk about the average values for the ensemble, as in statistical physics). Therefore, we limit the time to a certain limiting value T of order 1 ( T ~ 1 ): T B , where B > 0 is a conh h h stant. (In general, we say that the quantity u = u (h) in a DQM is of order hk (u ~ hk or u = O(hk)), if u ≤ Chk . And u = u (h) is exactly of the order hk (u ~ = hk), if chk ≤ u ≤ Chk for some constants C, c > 0).
Let us now consider the problem: it is required to experimentally determine the
location of the point at which a given point of the phase space (x; p) passes under the
action of dynamical system (
on the usual scale in the neighborhood of the point (x; p)
(see (
Definition 2. The Markov deviation Z (z) is a smooth vector field on phase space such that 1)
Z ( z) B
( B 0 ) is a constant of dynamical system (
for all points z of the phase space); 2) for any initial point z0 = z(t0) on the phase curve z(t) of the dynamical system
t0 t0
where e(t) is the unit normal vector of the closed phase curve at the point z(t),
B 0 is the constant of dynamical system (
Property (
If instead of a given time limit T ~ 1 we take T1 ~ 1 ( T1 T ), then we obtain
h h
another Markov deviation; similarly, when replacing the zero point in time. So the
Markov deviation is a smooth vector field that depends on the parameters; further, it
can be assumed to be a general view field.
(
The physical meaning of the eigenvalues is that these are all values of energy that can be the result of reliable, i.e. the most accurate measurement (ideally of the order of h2). But as a result of the most accurate experiments, as we have seen, in reality the dynamics is studied not of the diffeomorphism G, but of its perturbation G = G + Z . Let J = J (z) be a given smooth function on phase space. We can interpret it as the Hamiltonian (energy in the phase space) or as a function of truth (0 J (z) 1), equal to 1 on the true trajectory and 0 outside some neighborhood of it. Given the irremovable errors of the Markov deviation, the discreteness of the measurement process and its limited time, we arrive at the maximum number of the most accurate measurements 1 Nt J (G i z) , where z is the point of phase space, Z is the general view
Nt i=0 Markov deviation, G = G + Z is a diffeomorphism, Nt is the maximum number of significantly different measurements over time t ≤ T.
In terms of meaning the eigenvalue of the spectrum is associated with some pure
stationary state of the dynamical system: any reliable measurement in this state leads
to an acceptable error (ideally with a maximum accuracy of the order of h2) and only
to this value. But in a DQM any point in the phase space is always known with the
irremovable error of order h (see (
Definition 3. Let G = G + Z , where Z is a general view Markov deviation; Nt is the number of all iterations of the diffeomorphism in time t; α is a real number. Let D = Dαh be the set of points z of the phase space such that for all sufficiently large t < T ~ = 1 h 1 Nt J (G i z) − Bh 2 ,
Nt i=0
where B is a constant. Then, if for any Z of a general view and sufficiently small h,
the set Dαh contains a ball with a diameter of exactly the order of h, then α will be
called the eigenvalue of the DQM for dynamical system (
Thus, all points of a DQM spectrum are formally determined only with an accuracy of the order of h2, but this corresponds precisely to their meaning. By definition, the domain Dαh is an open G - invariant subset of the phase space.
So, to define DQM means to set: 1) the Markov process in accordance with Definition 1; 2) the Markov deviation Z of general view or, what is the same, diffeomorphism G = G + Z in accordance with Definition 2.
Consider the two–dimensional dynamical system (
Proposition 1. The DQM eigenvalues of the given dynamical system, with accuracy of the order h2, are equal to the values of J (z) on the phase circles in Λ, the ρ length of which satisfies the Bohr - Sommerfeld condition 1 p( x) dx = I = h(n + ) 2 and only they. 3
Consider the Schrödinger equation t
2
= H ( ) = − h2
x2 + U (x)
with the quasiperiodic potential U(x) = cos x + ε ∙ cos λx (ε, λ > 0). For this
Schrödinger equation we construct its dynamic quantum model (DQM), i.e. in accordance
with Definition 1, the perturbation of the corresponding classical system by the
Markov process. There is a simple connection between these Markov processes and the
quasisolutions of the Schrödinger equation. The connection between the Schrödinger
equation and the corresponding DQM is based on a following modification of the
traditional asymptotic expansion of solution (
i S ( x,t )
Lemma 1. Let (x, t) = (x, t) e h
is some quasisolution (
2 = H ( ) = − h2
x2 + U (x,t) + O(h2 ) , where (x,t) =h (x,t) and S(x, t) = Sh (x, t) are real-valued. Then S (x, t) t
+ S (xx, t) 2 + U (x) − h 2S (xx2, t) (x, t) −
− h (xt,t) + 2 S (xx,t) (xx,t) − h 2x(x2,t) = O(h2 ) . (
Now we show how the DQM of the Schrödinger equation is constructed from its
two-dimensional classical analogue
dx = p dp = − dU . (
According to section 2.3 the DQM of the Schrödinger equation is determined by
1) the smooth dynamics from (
1) Consider the first two terms of the asymptotic expansion of the solution S(x, t)
of equation (
S(x, t) = S0 (x, t) + hS1(x, t) + (h2 ) . (
Substituting this expansion into (
S0 (x, t) 2 + + U (x) = 0
x
On the trajectory (
t S0 (x, t) 2 + U (x) = . x S0 ( x, t) =
x
The velocity on γ is
− U (x) = р0(х) regardless of t.
Assuming that S1(x) is also independent of t, we find from (
x
Now the smooth DQM dynamics corresponding to the perturbed Hamilton - Jacobi
equation (
2) DQM also includes stochastic disturbance, “white noise” in the configuration
space. If each point x0, in accordance with the smooth dynamics of the DQM, moves
t0+t S
along the trajectory (
This equation is equivalent to (
Let H (γ) be the energy value on the trajectory (
Lemma 2. For all sufficiently small h, the eigenvalue α of equation (
I(γ) = h(n + 1 ) (n = 0, 1, …). (13)
2
Proof. Suppose that, on a closed trajectory (
+ U (x) = = H ( ) . As the initial x condition for S0(x, t) we take
x S0 ( x,0) = p0 ( y)dy , (14)
x0 where р0(y) = − U ( y) is velocity on γ, р0(x0) = 0, i.e. x0 is the abscissa of the turning point on γ. Then
S0 (x, t) = S0 (x, 0) − t + kt I ,
(15)
where kt is the number of turns in time t, πI is the ρ - length by γ between turning
points. And for S(x, t) = S0 (x,t) + hS1(x,t) , where in accordance with the construction
of DQM S1(x, t) = S1(x) from (
We assume that the velocity function р0(x) on the closed curve γ without loss of generality is analytic with an accuracy of the order of h2, otherwise approximating р0(x) on γ analytic with this accuracy. Then outside the neighborhood of the turning points this is also true for S0 (x, t) , S1(x), and in the neighborhood of the turning S(x, t) = S0 (x, t) + hS1(x, t) = S (x,0) − t + kt (I + hi 2 ) , where, as in (17), kt is the number of turns in time t.
By definition, DQM also includes a stochastic perturbation defined on γ by
equation (
x x − 2 x2 = O(h) ,
which is equivalent to this option (
S ( x, t) 0 ( x) + 2 x x
2 0 ( x)
− h x2
= O(h) .
point S1(x) we can continue analytically. When turning, the sign of S0 ( x, t) =
x
р0(х) changes, which in view of (
Now replace h by hi in (18), (17) and (16). Then 0 ( x) S ( x, t) 0 ( x)
+ 2 − ih t x x (x, t) = 0 ( x) e hi S ( x,t ) = 0 ( x) e hi (S ( x)−t ) ( S (x, t) t + S (x, t) 2 + U (x) − hi 2S (x, t) )φ(x, t) –
x x2 pair (α, 0 ( x) e hi S ( x) ) is quasisolution (quasimode) of the stationary Schrödinger equation α ψ = H ψ with a small parameter h2. It follows that α is an eigenvalue of the operator H with order accuracy h2.
Indeed, assuming that d is the distance from α to the spectrum of the operator H, ║ ║ is the norm in L2 , and R = ( E − H )−1 is resolvent H, we obtain for ψ = 0 ( x) e hi S ( x) in virtue of (25) , where B > 0, whence d ≤ B h2 and then the lemma follows from the Weyl criterion, QED.
Proposition 2. For Schrödinger equation (
Proof. At ε = 0, the potential U(x) = cos x is periodic. In this case, in the region
H(γ) ≤ 1 of the phase space of the dynamical system (
As increases from zero, the periodicity of the potential U(x) disappears, on the interval [2k; 2 (k + 1)] it has the form
U (x) = cos(x) + cos(2k + x) (x [0;2 ]). (26) If λ is rational and λ = p , then there are no more than q different such potentials q and for each of them the spectrum is discrete. For irrational λ, the points 2kλ modulo 2 everywhere densely fill the segment [0; 2] and their closure coincides with the segment.
Consider on the segment [0; 2] dynamic system (
As for potential U dI(H , , ) 0 at ε = 0, then this is also true for
suffidH
ciently small ε > 0. Let the inverse function f(I) = f(I, , ε) associates the ρ - length
I of a closed trajectory with its energy level H. According to Lemma 2, for I =
h(n + 1 ) (n = 0, 1, …) all this H = f(I, , ε) and there only are the eigenvalues (
The length of the segment KI smoothly depends on ε, and for sufficiently small h and ε of order h it increases approximately linearly with increasing ε. Therefore, with increasing ε, the union of the segments KI over all I = h(n + 1 ) (n = 0, 1, …) will 2 cover all values of energy in the field of oscillatory movements.
All the above considerations remain valid with a further increase in ε, and therefore the spectral picture does not change, only the range of vibrational motions expands. As ε → ∞, this region occupies the entire phase space. Moreover, in the spectral pattern in this region, a process occurs that is opposite to what was when ε changed from zero to h. Namely, the point spectrum narrows to segments converging in the limit to discrete points corresponding to phase curves with ρ – length I = h(n + 1 ) (n = 0, 2
At ε → ∞, i.e. ~ QED. 4
Conclusion ih
~ = − h t 2 2 ~x2
+ (~ cos x + cos x) .
→ 0 we back to the periodic case with potential U(x) = cos x,
1, …) for potential cos λx. This follows from the symmetry of the plots 0 < ε < h and
1 h < ε < ∞: divide equation (
But the definition of DQM is not formally related to Hamiltonian systems; it is defined for any ordinary differential equation or any diffeomorphism on any smooth Riemannian manifold. DQM is defined and constructed universally for both Hamiltonian systems and systems with the truth function. As a result, for example, the point of the DQM spectrum is interpreted exactly as the average value of truth from approximate logical conclusions. Quantization with the Bohr-Sommerfeld condition also extends to systems with a truth function.
Hopefully this will allow a new approach to the problems of dynamical systems that using logical operations: algorithms, theorems, software applications.