Anderson localization for fractional Laplacians
$begingroup$
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 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
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add a comment |
$begingroup$
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 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
$endgroup$
$begingroup$
I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597.
$endgroup$
– Mateusz Kwaśnicki
Dec 14 '18 at 20:06
$begingroup$
@MateuszKwaśnicki: thank you for this reference.
$endgroup$
– Abdelmalek Abdesselam
Dec 18 '18 at 11:56
add a comment |
$begingroup$
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 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
$endgroup$
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 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
asked Nov 24 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
asked Nov 24 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
asked Nov 24 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
asked Nov 24 '18 at 20:19
Abdelmalek AbdesselamAbdelmalek Abdesselam
10.9k12768
10.9k12768
$begingroup$
I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597.
$endgroup$
– Mateusz Kwaśnicki
Dec 14 '18 at 20:06
$begingroup$
@MateuszKwaśnicki: thank you for this reference.
$endgroup$
– Abdelmalek Abdesselam
Dec 18 '18 at 11:56
add a comment |
$begingroup$
I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597.
$endgroup$
– Mateusz Kwaśnicki
Dec 14 '18 at 20:06
$begingroup$
@MateuszKwaśnicki: thank you for this reference.
$endgroup$
– Abdelmalek Abdesselam
Dec 18 '18 at 11:56
$begingroup$
I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597.
$endgroup$
– Mateusz Kwaśnicki
Dec 14 '18 at 20:06
$begingroup$
I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597.
$endgroup$
– Mateusz Kwaśnicki
Dec 14 '18 at 20:06
$begingroup$
@MateuszKwaśnicki: thank you for this reference.
$endgroup$
– Abdelmalek Abdesselam
Dec 18 '18 at 11:56
$begingroup$
@MateuszKwaśnicki: thank you for this reference.
$endgroup$
– Abdelmalek Abdesselam
Dec 18 '18 at 11:56
add a comment |
1 Answer
1
active
oldest
votes
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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 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
$endgroup$
1
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
$endgroup$
1
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
add a comment |
$begingroup$
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 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
$endgroup$
1
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
add a comment |
$begingroup$
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 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
$endgroup$
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 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
answered Nov 24 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
answered Nov 24 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
answered Nov 24 '18 at 20:49
Carlo BeenakkerCarlo Beenakker
74.2k9169276
74.2k9169276
1
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
add a comment |
1
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
1
1
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
$begingroup$
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.
$endgroup$
– Abdelmalek Abdesselam
Nov 24 '18 at 20:56
add a comment |
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$begingroup$
I remember Kamil Kaleta and Katarzyna Pietruska-Pałuba did some research on non-local operators with random potentials. Not sure if this is relevant to your question (and certainly not on the physics side), but anyway you may like to have a look at their arXiv:1601.05597.
$endgroup$
– Mateusz Kwaśnicki
Dec 14 '18 at 20:06
$begingroup$
@MateuszKwaśnicki: thank you for this reference.
$endgroup$
– Abdelmalek Abdesselam
Dec 18 '18 at 11:56