Eigenvectors of the 1-dimensional critical random Schrödinger operator
Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptio...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Rifkind, Ben [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Anmerkung: |
© Springer International Publishing AG, part of Springer Nature 2018 |
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| Übergeordnetes Werk: |
Enthalten in: Geometric and functional analysis - Springer International Publishing, 1991, 28(2018), 5 vom: 14. Aug., Seite 1394-1419 |
|---|---|
| Übergeordnetes Werk: |
volume:28 ; year:2018 ; number:5 ; day:14 ; month:08 ; pages:1394-1419 |
| Links: |
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| DOI / URN: |
10.1007/s00039-018-0460-0 |
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OLC2060222257 |
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| 520 | |a Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. | ||
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10.1007/s00039-018-0460-0 doi (DE-627)OLC2060222257 (DE-He213)s00039-018-0460-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Rifkind, Ben verfasserin aut Eigenvectors of the 1-dimensional critical random Schrödinger operator 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. Virág, Bálint aut Enthalten in Geometric and functional analysis Springer International Publishing, 1991 28(2018), 5 vom: 14. Aug., Seite 1394-1419 (DE-627)130964344 (DE-600)1067164-X (DE-576)028038169 1016-443X nnns volume:28 year:2018 number:5 day:14 month:08 pages:1394-1419 https://doi.org/10.1007/s00039-018-0460-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4311 AR 28 2018 5 14 08 1394-1419 |
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10.1007/s00039-018-0460-0 doi (DE-627)OLC2060222257 (DE-He213)s00039-018-0460-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Rifkind, Ben verfasserin aut Eigenvectors of the 1-dimensional critical random Schrödinger operator 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. Virág, Bálint aut Enthalten in Geometric and functional analysis Springer International Publishing, 1991 28(2018), 5 vom: 14. Aug., Seite 1394-1419 (DE-627)130964344 (DE-600)1067164-X (DE-576)028038169 1016-443X nnns volume:28 year:2018 number:5 day:14 month:08 pages:1394-1419 https://doi.org/10.1007/s00039-018-0460-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4311 AR 28 2018 5 14 08 1394-1419 |
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10.1007/s00039-018-0460-0 doi (DE-627)OLC2060222257 (DE-He213)s00039-018-0460-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Rifkind, Ben verfasserin aut Eigenvectors of the 1-dimensional critical random Schrödinger operator 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. Virág, Bálint aut Enthalten in Geometric and functional analysis Springer International Publishing, 1991 28(2018), 5 vom: 14. Aug., Seite 1394-1419 (DE-627)130964344 (DE-600)1067164-X (DE-576)028038169 1016-443X nnns volume:28 year:2018 number:5 day:14 month:08 pages:1394-1419 https://doi.org/10.1007/s00039-018-0460-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4311 AR 28 2018 5 14 08 1394-1419 |
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10.1007/s00039-018-0460-0 doi (DE-627)OLC2060222257 (DE-He213)s00039-018-0460-0-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Rifkind, Ben verfasserin aut Eigenvectors of the 1-dimensional critical random Schrödinger operator 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. Virág, Bálint aut Enthalten in Geometric and functional analysis Springer International Publishing, 1991 28(2018), 5 vom: 14. Aug., Seite 1394-1419 (DE-627)130964344 (DE-600)1067164-X (DE-576)028038169 1016-443X nnns volume:28 year:2018 number:5 day:14 month:08 pages:1394-1419 https://doi.org/10.1007/s00039-018-0460-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4311 AR 28 2018 5 14 08 1394-1419 |
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Eigenvectors of the 1-dimensional critical random Schrödinger operator |
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Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. © Springer International Publishing AG, part of Springer Nature 2018 |
| abstractGer |
Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. © Springer International Publishing AG, part of Springer Nature 2018 |
| abstract_unstemmed |
Abstract The purpose of this paper is to understand in more detail the shape of the eigenvectors of the random Schrödinger operator $${H = \Delta + V}$$ on $${\ell^2(\mathbb{Z})}$$. Here $${\Delta}$$ is the discrete Laplacian and V is a random potential. It is well known that under certain assumptions on V the spectrum of this operator is pure point and its eigenvectors are exponentially localized; a phenomenon known as Anderson Localization. We restrict the operator to $${\mathbb{Z}_n}$$ and consider the critical model, (Hnψ)ℓ=ψℓ-1,n+ψℓ+1,n+vℓ,nψℓ,ψ0=ψn+1=0,$$(H_n \psi)_\ell =\psi_{\ell -1,n} +\psi_{\ell +1,n} + v_{\ell,n} \psi_\ell , \quad \psi_0 = \psi_{n+1}=0,$$with vk are independent random variables with mean 0 and variance $${\sigma^2/n}$$. We show that the scaling limit of the shape of a uniformly chosen eigenvector of Hn is exp-|t-U|4+Zt-U2,$${\rm exp} \left ( - \frac{|t-U|}{4} + \frac{\mathcal{Z}_{t-U}}{\sqrt{2}} \right ), $$where U is uniform on [0,1] and $${\mathcal{Z}}$$ is an independent two sided Brownian motion started from 0. © Springer International Publishing AG, part of Springer Nature 2018 |
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| title_short |
Eigenvectors of the 1-dimensional critical random Schrödinger operator |
| url |
https://doi.org/10.1007/s00039-018-0460-0 |
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Virág, Bálint |
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Virág, Bálint |
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| doi_str |
10.1007/s00039-018-0460-0 |
| up_date |
2024-07-04T00:50:12.302Z |
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