Anderson localization for fractional Laplacians
up vote
5
down vote
favorite
There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(mathbb{Z}^d)$ such as
$$
-Delta+lambda V
$$
where $Delta$ is the discrete lattice Laplacian and the potential $V$ is random say given by a vector $(V_{mathbf{x}})_{mathbf{x}inmathbb{Z}^d}$ of iid standard normal variables. The constant $lambda$ is the strength of disorder.
Did anyone study similar random operators
$$
(-Delta)^{alpha}+lambda V
$$
with a fractional Laplacian?
I am in particular interested in references from the physics literature which provide some heuristics, e.g., a scaling theory à la Abrahams et al.
reference-request mp.mathematical-physics schrodinger-operators fractional-calculus
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
add a comment |
up vote
5
down vote
favorite
There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(mathbb{Z}^d)$ such as
$$
-Delta+lambda V
$$
where $Delta$ is the discrete lattice Laplacian and the potential $V$ is random say given by a vector $(V_{mathbf{x}})_{mathbf{x}inmathbb{Z}^d}$ of iid standard normal variables. The constant $lambda$ is the strength of disorder.
Did anyone study similar random operators
$$
(-Delta)^{alpha}+lambda V
$$
with a fractional Laplacian?
I am in particular interested in references from the physics literature which provide some heuristics, e.g., a scaling theory à la Abrahams et al.
reference-request mp.mathematical-physics schrodinger-operators fractional-calculus
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
add a comment |
up vote
5
down vote
favorite
up vote
5
down vote
favorite
There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(mathbb{Z}^d)$ such as
$$
-Delta+lambda V
$$
where $Delta$ is the discrete lattice Laplacian and the potential $V$ is random say given by a vector $(V_{mathbf{x}})_{mathbf{x}inmathbb{Z}^d}$ of iid standard normal variables. The constant $lambda$ is the strength of disorder.
Did anyone study similar random operators
$$
(-Delta)^{alpha}+lambda V
$$
with a fractional Laplacian?
I am in particular interested in references from the physics literature which provide some heuristics, e.g., a scaling theory à la Abrahams et al.
reference-request mp.mathematical-physics schrodinger-operators fractional-calculus
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(mathbb{Z}^d)$ such as
$$
-Delta+lambda V
$$
where $Delta$ is the discrete lattice Laplacian and the potential $V$ is random say given by a vector $(V_{mathbf{x}})_{mathbf{x}inmathbb{Z}^d}$ of iid standard normal variables. The constant $lambda$ is the strength of disorder.
Did anyone study similar random operators
$$
(-Delta)^{alpha}+lambda V
$$
with a fractional Laplacian?
I am in particular interested in references from the physics literature which provide some heuristics, e.g., a scaling theory à la Abrahams et al.
reference-request mp.mathematical-physics schrodinger-operators fractional-calculus
reference-request mp.mathematical-physics schrodinger-operators fractional-calculus
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
asked Nov 24 at 20:19
Abdelmalek Abdesselam
10.7k12667
10.7k12667
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
Since fractional Laplacians describe diffusion on a fractal, for a physics application I would focus on Anderson localization on a fractal. The scaling theory mentioned in the OP has been applied to a fractal in An attractive critical point from weak antilocalization on fractals.
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
1
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Since fractional Laplacians describe diffusion on a fractal, for a physics application I would focus on Anderson localization on a fractal. The scaling theory mentioned in the OP has been applied to a fractal in An attractive critical point from weak antilocalization on fractals.
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
1
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
add a comment |
up vote
2
down vote
Since fractional Laplacians describe diffusion on a fractal, for a physics application I would focus on Anderson localization on a fractal. The scaling theory mentioned in the OP has been applied to a fractal in An attractive critical point from weak antilocalization on fractals.
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
1
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
add a comment |
up vote
2
down vote
up vote
2
down vote
Since fractional Laplacians describe diffusion on a fractal, for a physics application I would focus on Anderson localization on a fractal. The scaling theory mentioned in the OP has been applied to a fractal in An attractive critical point from weak antilocalization on fractals.
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
Since fractional Laplacians describe diffusion on a fractal, for a physics application I would focus on Anderson localization on a fractal. The scaling theory mentioned in the OP has been applied to a fractal in An attractive critical point from weak antilocalization on fractals.
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
answered Nov 24 at 20:49
Carlo Beenakker
71.7k9160267
71.7k9160267
1
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
add a comment |
1
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
1
1
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
Thanks for the reference. Although, I am not sure about the validity of the equivalence between fractional Laplacian on the square lattice versus ordinary Laplacian on fractal lattices. I would prefer references on the former.
– Abdelmalek Abdesselam
Nov 24 at 20:56
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f316128%2fanderson-localization-for-fractional-laplacians%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
