2
$\begingroup$

What is the physics interpretation of Sobolev space? $H^{s,p}:=\left\{u\in L^p(\mathbb{R}^n):\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))\in L^p(\mathbb{R}^n)\right\}$, $s\geq 0,\, 1<p<\infty$ (Sobolev space) For example, if $n=3,\,s=2,\,p=2$. Is there an interpretation?

Ask this, to find out if the equation $\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))=g$ with $g\in L^2(\mathbb{R}^n)$ has any physical interpretation or application.

$\endgroup$
  • 3
    $\begingroup$ This seems similarly ill-defined to asking "What is the physics interpretation of $L^2(\mathbb{R})$?". There are many parts of physics where square-integrable functions play a role and that space is relevant, but none of these is really "the physics interpretation" of it - without physical context given, that space is just a mathematical construct without any inherent physical meaning. $\endgroup$ – ACuriousMind Jun 13 at 13:19
5
$\begingroup$

The (non-homogeneous) Sobolev spaces $H^{p,s}$ are the subspaces of $L^p$ of functions that admit $s$ weak derivatives belonging to the same Lebesgue space, where a weak derivative is a derivative in the sense of distributions.

The foremost ones of physical significance are the ones $H^s=H^{2,s}$ where the base space is $L^2$. This is due to the fact that $L^2$ is the prototypical Hilbert space for nonrelativistic quantum mechanics.

As a concrete example, $H^2$ is the natural domain of definition, and self-adjointness, of the Laplace operator $-\Delta$, that is the kinetic energy of a quantum particle.

Another example may be given by nonlinear Schrödinger equations of the type $$i\partial_t \psi = -\Delta \psi + \lvert \psi\rvert^{p-1} \psi \; .$$ Such equations are crucial to describe the effective behavior of condensed matter systems, such as the motion of atoms in a BEC. The existence and uniqueness of solutions to such equations is typically investigated and proved in Sobolev spaces $H^s$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ A distribution possesses a weak derivatives of any order. I think that by "admits" @yuggib means that the $s$-order weak derivatives of elements of $H^{p,s}$ are still elements of $L^p$. Is this correct? $\endgroup$ – mike stone Jun 13 at 13:02
  • $\begingroup$ @mikestone Yes, this is what I mean, I'll edit for clarity $\endgroup$ – yuggib Jun 13 at 13:08
2
$\begingroup$

By stretching the question just a little bit, there is in fact a very physical interpretation. For a Euclidean QFT with say a scalar field $\phi$ in $d$ spacetime dimensions, the latter is typically (when Nelson-Symanzik positivity holds in addition to Osterwalder-Schrader positivity) realized as a random Schwartz distribution in $\mathscr{S}'(\mathbb{R}^d)$. One may then ask for which (weighted inhomogeneous) Besov spaces $B_{p,q}^{s}$ is $\phi$ almost surely in that space. The typical answer is: for all $s<-\Delta_{\phi}$ where $\Delta_{\phi}$ is an important physical quantity, namely the scaling dimension of the field $\phi$. This is just the distributional generalization of the Kolmogorov-Chentsov Theorem.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.