Figuring symmetries of a differential operator from its eigenfunctions

In summary, the derivative operator ##\frac{d}{dx}## has translational invariance while the theta operator ##x\frac{d}{dx}## has scale invariance. Both operators have eigenfunctions and their respective eigenvalues are preserved under the operators. It is possible to construct a sequence of operators and their eigenfunctions with similar properties. These symmetries are associated with conservation laws and can be used to determine the "finite" version of the transformation from its infinitesimal counterpart.
  • #1
JPaquim
34
0
So, I understand that the derivative operator, [itex]D=\frac{d}{dx}[/itex] has translational invariance, that is: [itex]x \rightarrow x - x_0[/itex], and its eigenfunctions are [itex]e^{\lambda t}[/itex]. Analogously, the theta operator [itex]\theta=x\frac{d}{dx}[/itex] is invariant under scalings, that is [itex]x \rightarrow \alpha x[/itex], and its eigenfunctions are [itex]x^\lambda[/itex]. Taking logarithms and exponentials, I have constructed a sequence of operators and their respective eigenfunctions, all with the property that [itex]\{L(\frac{d}{dx})\}f^\lambda(x)=\lambda f^\lambda(x)[/itex]. I've taken a picture and attached it to this post.

My guess is that associated with every single one of these operators is some symmetry, some sort of coordinate transformation [itex]x \rightarrow f(x)[/itex] under which the operator is invariant. For the [itex]x\log x \frac{d}{dx}[/itex] operator, its invariant under [itex]x \rightarrow x^k[/itex], by inspection. How can I figure out what sort of symmetry a given operator has, given its eigenfunctions?

Physically, symmetries are associated with conservation laws. For a system whose differential equations are governed by this sort of differential operators, what sort of conserved quantities should I expect?
 

Attachments

  • IMG_0596.jpg
    IMG_0596.jpg
    16.5 KB · Views: 392
Mathematics news on Phys.org
  • #2
##\frac{d}{dx}## is associated with translational symmetry. An infinitesimal translation is produced by acting on a function with the operator ##1 + \epsilon \frac{d}{dx}##, with ##\epsilon## infinitesimal.

##x \frac{d}{dx}## is associated with scale invariance. An infinitesimal rescaling is produced by acting on a function with the operator ##1 + \epsilon x \frac{d}{dx}##, with ##\epsilon## infinitesimal.

Presumably your other operators ##O## can be associated with symmetry transformations with an infinitesimal transformation being implemented by ##1 + \epsilon O##?
 
  • #3
Ok, I agree with you. How can I figure out the "finite" version of the transformation from its infinitesimal counterpart?
 

1. What are symmetries in the context of a differential operator?

Symmetries in the context of a differential operator refer to transformations that leave the operator unchanged. These transformations can be rotations, reflections, translations, or other types of transformations.

2. How can eigenfunctions be used to determine the symmetries of a differential operator?

Eigenfunctions are special functions that satisfy the differential equation of the operator and have unique properties. By analyzing the behavior of these eigenfunctions under different transformations, we can determine the symmetries of the operator.

3. What is the significance of figuring out the symmetries of a differential operator?

Knowing the symmetries of a differential operator can provide valuable insights into the behavior and properties of the operator. It can also help in solving differential equations and understanding the underlying physical or mathematical principles.

4. What techniques are commonly used to figure out the symmetries of a differential operator?

There are various techniques that can be used, such as the Lie algebra method, the symmetry reduction method, and the group theory approach. These techniques involve analyzing the properties of the eigenfunctions and the operator itself.

5. Can the symmetries of a differential operator change depending on the boundary conditions?

Yes, the symmetries of a differential operator can change depending on the boundary conditions imposed on the system. Different boundary conditions can lead to different sets of eigenfunctions and thus, different symmetries of the operator.

Similar threads

Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
1K
  • Quantum Physics
Replies
14
Views
829
Replies
9
Views
2K
  • Differential Equations
Replies
1
Views
1K
Replies
7
Views
3K
  • Special and General Relativity
Replies
3
Views
2K
Replies
1
Views
1K
Replies
14
Views
1K
Replies
3
Views
856
Back
Top