Skip to main content# Resolution of the singularity of the ODE x'=x^2 when the initial x has uncertainties

# Abstract

**General picture**

# State-of-the-art

# Resolution of the singularity using the uncertainties in x

# Conjecture

We show that the classical initial value problem x'=x^2 has a continuous solution for all time, when the initial x has any finite amount of uncertainties (which is unavoidable in engineering). This is in stark contrast with the singular solution when x has no uncertainties.

Published onApr 13, 2023

Resolution of the singularity of the ODE x'=x^2 when the initial x has uncertainties

We show that the classical initial value problem x'=x^2 has a continuous solution for all time, when the initial x has any finite amount of uncertainties (which is unavoidable in engineering). This is in stark contrast with the singular solution when x has no uncertainties.

Consider a sample space defined by the real line. Since we can always define a wave-function by taking the square-root of the probability measure, the Koopman-von Neumann version of classical statistical mechanics[1][2][3][4] defines classical statistical mechanics as a particular case of quantum mechanics where the algebra of observable operators is necessarily commutative (because the time-evolution is deterministic).

However, it is well known that for two unbounded self-adjoint operators, the fact that they commute does not imply that their corresponding spectral measures commute[5], then we would say that they *strongly* commute. Thus in principle, it is possible to have a classical dynamical system defined by an *ordinary differential equation* where the unitary time-evolution is non-deterministic, that is, the algebra of observable operators is non-commutative. We will not study this possibility here, this is consistent with the fact that in a continuous standard probability space we can not neglect uncertainties, for that we would need to work in the larger space of finitely additive measures[6], which has very different properties than a standard probability space. Note that such solutions non-deterministic solutions do not appear if at the initial time, there is a joint probability distribution for all the random variables appearing in the ordinary differential equation, which is a common assumption.

Since the real line is a simply connected group (for addition), then by Bargmann’s theorem any strongly continuous projective representation of the real line is induced by some strongly continuous representation of the real line on the corresponding Hilbert space[7][8]. Then, Stone’s theorem implies that such representation on the Hilbert space is the exponential of a time-independent Hamiltonian[7][8]. Thus, if there is a solution in a standard probability space of a differential equation without singularities for all times (including negative times), then there is a quantum Hamiltonian corresponding to such solution which is time-independent. Moreover, if such solution is deterministic then the fact that there are more Hamiltonians that generate the same differential equation does not change the unique solution, up to initial conditions: suppose that

Moreover, any solution in a standard probability space is given by a time-dependent self-adjoint Hamiltonian in a larger probability space, then the singularities in a standard probability space are consequence of a time-dependent Hamiltonian which is non-integrable. But since any time-dependent Hamiltonian can be converted into a time-independent Hamiltonian in an even larger sample space, then we can always resolve the singularities in a larger sample space which better defines the dynamical system. Thus, solutions without singularities for all times (including negative times) defined by time-independent Hamiltonians are expressive enough, even to deal with singularities at the cost of choosing a larger sample space.

Consider the differential equation:

$x(t)=\frac{x(0)}{1-t\ x(0)} \tag{1}$

The general solution has a singularity at

When considering uncertainties, if we assume that the solution Equation 1 exists for some

We could think that a unitary time-evolution is somehow too demanding, that only with restrictions to the allowed initial probability measures we can find all solutions to the differential equation defined above. But as a matter of principle that should not be correct since we always work with such restrictions under the assumption that we will somehow be able to recover a corresponding result in the space of probability measures, otherwise such result cannot be tested experimentally or even in most numerical simulations (due to approximation uncertainties). Indeed, if we consider the space

Changing the sample space, effectively changes the equation, since

The quantum Hamiltonian operator corresponding to *strongly *commute for all times. If they *strongly *commute for all times, then they can be simultaneously diagonalized and there is a deterministic solution without singularities. This would be in stark contrast with the fact that there are no deterministic solutions without singularities, when the uncertainty is neglected[9].

The unitary operator:

$U(x,p)=\frac{e^{-i \frac{p}{x}}}{\sqrt{2\pi} x } \tag{2}$

It diagonalizes the Hamiltonian, that is

$\begin{aligned}
&e^{-i H t}\{\psi\}(x)=\int\limits_{-\infty}^{+\infty}dp\int\limits_{-\infty}^{+\infty}dy\ U(x,p)e^{-ip t} U^*(y,p)\psi(y)\\
&=\int\limits_{-\infty}^{+\infty}\frac{dy}{2\pi x y}\ 2\pi\delta\biggl(t+\frac{1}{x}-\frac{1}{y}\biggr)\psi(y)\\
&=\int\limits_{-\infty}^{+\infty}\frac{dw}{x w}\ \delta\biggl(t+\frac{1}{x}-w\biggr)\psi\biggl(\frac{1}{w}\biggr)=\frac{1}{x t+1}\psi\biggl(\frac{x}{x t+1}\biggr)
\end{aligned} \tag{3}$

Where we did the change of variables

$\begin{aligned}
&\rho\{\psi\}(x)=\rho(x)\psi(x)\\
&(e^{-i H t}\rho\ e^{i H t})\{\psi\}(x)=\\
&=\frac{1}{x t+1}\int\limits_{-\infty}^{+\infty} \frac{dy}{1-y t}\ \delta\biggl(\frac{x}{x t+1}-y\biggr)\rho(y)\psi\biggl(\frac{y}{1-y t}\biggr)=\\
&=\rho\biggl(\frac{x}{x t+1}\biggr)\psi(x)
\end{aligned} \tag{4}$

A singularity at finite-time in the time-evolution of an initial value problem defined by an ordinary differential equation implies that the derivative in time of the corresponding wave-function diverges. In the case of x’=x^2, the coordinate goes instantly from

Since the derivative in time corresponds to the Hamiltonian (whose spectral measure is conserved by the time-evolution) then if the initial condition only involves eigenfunctions corresponding to a spectrum up to some maximum absolute value, there is no way in which the time-evolution will produce a divergent derivative in time of the wave-function. Such initial condition also makes sense from a physics point of view, where we usually have limited energy available when we apply physical transformations to a system; it also makes sense from the point of view of numerical calculations, where the Krylov subspace methods for computations of matrix exponentials is guaranteed to converge for any wave-function defined in an Hilbert space[13][14] (even when the Hamiltonian is unbounded), that is, one where an initial condition with infinite energy has null measure (otherwise the well known phenomenon aliasing occurs).

Thus, the sample space of our initial system is usually not the coordinate space, since it verifies an energy limit. We can relate such initial sample space to the coordinate space through a physical transformation, which is only hypothetical, because it would require the availability of an arbitrarily large amount of energy. That is, we can only guess what the coordinate x would be (with an uncertainty) if we would do the necessary transformation, but never really do such transformation and thus never measure the coordinate.

Since the Hamiltonian differs from the translation in space by a change of variables

Note that since the probability is the predual of the algebra of events, the meaning of the time-evolution of the probability is to define what is the probability in the past which would produce the system in the present. Thus, the time-evolution does not make an object cross the would-be singularity, since only probabilities resulting from an hypothetical physical transformation (which never occurs because we do not have enough energy available) are transferred from some points in space to others. We need to look for what happens in the sample space of the real system (and not an hypothetical one), where the time-evolution is non-deterministic. The time-evolution would be deterministic in the coordinate space, but it is non-deterministic in the sample space which verifies an energy limit: that is the price to pay for resolving the singularity.

Note that the claim that limiting the energy of the initial condition suffices to solve the singularity is not by itself in conflict with the special relativity, because special relativity is expressed in Quantum Mechanics through an Hamiltonian that verifies extra conditions. By limiting the energy (Hamiltonian’s spectrum) of the initial condition we do not break such extra conditions, because the spectrum of the Hamiltonian verifies the same extra conditions as the Hamiltonian itself.

We can easily see that for instance, all time-independent polynomial quantum Hamiltonians such that the domain is the dense set of smooth functions with compact support, can be proved to be essentially self-adjoint[11][12], using