Differentials, pseudo-differentials, and locality

In summary, the claims 1,2, and 3 are all true. The pseudo-differential operator \sqrt{1-\partial_x^2} does not possess the locality property, but it is possible that it still has this property. However, the answer to the question posed is "no", as the instantaneous spreading caused by this operator is significantly faster than that caused by the ordinary differential operator \partial_x^2. It is also not obvious how to prove that the solution to the non-relativistic Schrödinger equation does not have a compact support for t>0, as there could be canceling in the integral. The spreading caused by this equation is faster than that of the heat equation, and even faster
  • #1
jostpuur
2,116
19
Claim 1:

If [itex]\psi(x,0)[/itex] has a compact support, and
[tex]
i\partial_t\psi(x,t) = -\partial_x^2 \psi(x,t),
[/tex]
then [itex]\psi(x,t)[/itex] does not have a compact support for any [itex]t>0[/itex].

Claim 2:

If [itex]\psi_1[/itex] and [itex]\psi_2[/itex] are the same in some environment of a point [itex]x_0[/itex], then
[tex]
\partial_x^2 \psi_1(x_0) = \partial_x^2 \psi_2(x_0)
[/tex]

Claim 3:

If [itex]\psi(x,0)[/itex] has a compact support, and
[tex]
i\partial_t\psi(x,t) = \sqrt{1 - \partial_x^2} \psi(x,t),
[/tex]
then [itex]\psi(x,t)[/itex] does not have a compact support for any [itex]t>0[/itex].

Question:

If [itex]\psi_1[/itex] and [itex]\psi_2[/itex] are the same in some environment of [itex]x_0[/itex], will it follow that
[tex]
\sqrt{1 - \partial_x^2}\psi_1(x_0) = \sqrt{1 - \partial_x^2}\psi_2(x_0)?
[/tex]

Thoughts:

According to my understanding, at least with some assumptions, the claims 1,2,3 are all true. If I had not mentioned the claims 1 and 2, some people might have answered to my question, that the pseudo-differential operator [itex]\sqrt{1 - \partial_x^2}[/itex] does not possesses the locality property I'm asking in the question, because we know that it is non-local in the sense of the claim 3. However, we know that the ordinary differential operator [itex]\partial_x^2[/itex] has the non-locality property in the sense of claim 1 too, and still it has the locality property in the sense of claim 2. So without better knowledge, it could be that the pseudo-differential operator has this locality property too. But what's the truth?
 
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  • #2
I think that the answer to my question is "no", and the instantaneous spreading caused by [tex]\sqrt{1 - \partial_x^2}[/tex] is significantly faster than the instantaneous spreading caused by [tex]\partial_x^2[/tex].

I just realized that I don't yet know how to prove that the non-relativistic Schrödinger equation causes instantaneous spreading. Here's the problem:

If we want to satisfy the heat equation

[tex]
\partial_t u(t,x) = \partial_x^2 u(t,x)
[/tex]

with some initial condition [itex]u(0,x)\geq 0[/itex] which has compact support, we can accomplish it with a formula

[tex]
u(t,x) = \frac{1}{\sqrt{4\pi t}}\int\limits_{-\infty}^{\infty} \exp\Big(-\frac{y^2}{4t}\Big) u(0,x-y)dy.
[/tex]

It can be seen from this formula that [itex]u(t,x)>0[/itex] for all [itex]t>0[/itex]. Hence the compact support is lost instantaneously.

But the take a closer look at the non-relativistic Schrödinger equation. We want

[tex]
i\partial_t \psi(t,x) = -\partial_x^2\psi(t,x)
[/tex]

with some initial condition [itex]\psi(0,x)[/itex] with compact support. The solution can be written as

[tex]
\psi(t,x) = \frac{1}{\sqrt{4\pi t}}\int\limits_{-\infty}^{\infty} \exp\Big(\frac{iy^2}{4t}\Big) \psi(0,x-y)dy.
[/tex]

But how do you prove that [itex]\psi(t,x)[/itex] does not have a compact support for [itex]t>0[/itex]? It's not so obvious now because there could be some canceling in the integral.

I'm interested in rigor comments only! (Or comments that at least attempt some rigor. I'm not sure if I'm perfectly rigorous either...).

I know some physicists like to say that "well look at the propagator. It is non-zero for arbitrarily large intervals, and hence the instantaneous spreading." You might as well try to insist that Fourier transforms (or inverse transforms) will never give functions with compact support, because the plane waves extend to infinities. Of course canceling can occur!

An interesting remark about the heat equation is that if we fix some [itex]x'[/itex] outside the original compact support, then the Taylor series

[tex]
u(0,x') + t \partial_t u(0,x') + \frac{1}{2!} t^2 \partial_t^2 u(0,x') + \cdots = 0
[/tex]

is identically zero, even though [itex]u(t,x')>0[/itex] for arbitrarily small [itex]t>0[/itex]. This means that the spreading is extremely slow, at least for small [itex]t[/itex].

The same slowness does not occur for relativistic Schrödinger equation. If [itex]\psi(0,x)[/itex] has a compact support, then [tex]\hat{\psi}(0,p)[/tex] is analytic. But

[tex]
i\partial_t\hat{\psi}(0,p) = \sqrt{1 + p^2}\hat{\psi}(0,p)
[/tex]

is not analytic because of the branch cuts at [itex]p=\pm i[/itex]. Hence [itex]\partial_t\psi(0,x)[/itex] does not have a compact support, and the instantaneous spreading is faster than with the heat equation.
 

1. What is the difference between differentials and pseudo-differentials?

Differentials are a mathematical concept used to describe the infinitesimal change in a variable. They are denoted by the symbol "d" and are often used in calculus to find derivatives and integrals. Pseudo-differentials, on the other hand, are a generalization of differentials that take into account non-smooth functions and non-Euclidean spaces. They are denoted by the symbol "∂" and are commonly used in partial differential equations.

2. How are differentials and pseudo-differentials used in physics?

Differentials and pseudo-differentials are used in physics to describe the behavior of physical systems and to solve equations that govern their motion. In classical mechanics, differentials are used to calculate the position, velocity, and acceleration of a particle, while pseudo-differentials are used in quantum mechanics to describe the wave functions of particles.

3. What is locality in relation to differentials and pseudo-differentials?

Locality refers to the principle that the behavior of a system at a given point depends only on the values of its variables at that point, and not on their values at other points. In the context of differentials and pseudo-differentials, this means that the outcome of a differential or pseudo-differential operation is determined solely by the values of the variables at the point of operation.

4. How do differentials and pseudo-differentials relate to the concept of smoothness?

Differentials and pseudo-differentials are closely related to the concept of smoothness, which refers to the degree of differentiability of a function. Differentials are used to describe smooth functions, while pseudo-differentials can handle non-smooth functions. In general, smoothness is important in the use of differentials and pseudo-differentials as it allows for the accurate analysis and prediction of physical systems.

5. Can differentials and pseudo-differentials be applied to non-linear systems?

Yes, differentials and pseudo-differentials can be applied to non-linear systems. In fact, they are often used in the study of non-linear systems as they allow for a more accurate and precise analysis of their behavior. The use of differentials and pseudo-differentials in non-linear systems is especially important in fields such as chaos theory and fluid dynamics.

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