### Abstract

In this note, it is shown that the stochastic Airy operator, which is the "Schrödinger operator" on the half-line whose potential term consists of Gaussian white noise plus a linear term tending to †∞, can naturally be defined as a generalized Sturm-Liouville operator and that it is self-adjoint and has purely discrete spectrum with probability one. Thus "stochastic Airy spectrum" of Ramírez, Rider and Virág is the spectrum of an operator in the ordinary sense of the word.

Original language | English |
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Pages (from-to) | 695-711 |

Number of pages | 17 |

Journal | Markov Processes and Related Fields |

Volume | 21 |

Issue number | 3P |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Random Schrödinger operator
- Self-adjointness
- Sturm-Liouville operator

### ASJC Scopus subject areas

- Applied Mathematics
- Statistics and Probability

### Cite this

*Markov Processes and Related Fields*,

*21*(3P), 695-711.

**Definition and self-adjointness of the stochastic airy operator.** / Minami, Nariyuki.

Research output: Contribution to journal › Article

*Markov Processes and Related Fields*, vol. 21, no. 3P, pp. 695-711.

}

TY - JOUR

T1 - Definition and self-adjointness of the stochastic airy operator

AU - Minami, Nariyuki

PY - 2015

Y1 - 2015

N2 - In this note, it is shown that the stochastic Airy operator, which is the "Schrödinger operator" on the half-line whose potential term consists of Gaussian white noise plus a linear term tending to †∞, can naturally be defined as a generalized Sturm-Liouville operator and that it is self-adjoint and has purely discrete spectrum with probability one. Thus "stochastic Airy spectrum" of Ramírez, Rider and Virág is the spectrum of an operator in the ordinary sense of the word.

AB - In this note, it is shown that the stochastic Airy operator, which is the "Schrödinger operator" on the half-line whose potential term consists of Gaussian white noise plus a linear term tending to †∞, can naturally be defined as a generalized Sturm-Liouville operator and that it is self-adjoint and has purely discrete spectrum with probability one. Thus "stochastic Airy spectrum" of Ramírez, Rider and Virág is the spectrum of an operator in the ordinary sense of the word.

KW - Random Schrödinger operator

KW - Self-adjointness

KW - Sturm-Liouville operator

UR - http://www.scopus.com/inward/record.url?scp=84960945514&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960945514&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84960945514

VL - 21

SP - 695

EP - 711

JO - Markov Processes and Related Fields

JF - Markov Processes and Related Fields

SN - 1024-2953

IS - 3P

ER -