### 抄録

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard FeynmanKac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L ^{p}-L ^{q} bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

元の言語 | 英語 |
---|---|

記事番号 | 1250013 |

ジャーナル | Reviews in Mathematical Physics |

巻 | 24 |

発行部数 | 6 |

DOI | |

出版物ステータス | 出版済み - 7 1 2012 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### これを引用

*Reviews in Mathematical Physics*,

*24*(6), [1250013]. https://doi.org/10.1142/S0129055X12500134

**Path integral representation for schrödinger operators with bernstein functions of the laplacian.** / Hiroshima, Fumio; Ichinose, Takashi; Lrinczi, József.

研究成果: ジャーナルへの寄稿 › 記事

*Reviews in Mathematical Physics*, 巻. 24, 番号 6, 1250013. https://doi.org/10.1142/S0129055X12500134

}

TY - JOUR

T1 - Path integral representation for schrödinger operators with bernstein functions of the laplacian

AU - Hiroshima, Fumio

AU - Ichinose, Takashi

AU - Lrinczi, József

PY - 2012/7/1

Y1 - 2012/7/1

N2 - Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard FeynmanKac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L p-L q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

AB - Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard FeynmanKac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an L p-L q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

UR - http://www.scopus.com/inward/record.url?scp=84862563972&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862563972&partnerID=8YFLogxK

U2 - 10.1142/S0129055X12500134

DO - 10.1142/S0129055X12500134

M3 - Article

VL - 24

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 6

M1 - 1250013

ER -