### Abstract

We study the spectrum of the Schrödinger operators with n × n matrix valued potentials on a finite interval subject to θ−periodic boundary conditions. For two such operators, corresponding to different values of θ, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to θ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a fixed θ.

Language | English (US) |
---|---|

Pages | 363-377 |

Number of pages | 15 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Differential operators
- Discrete spectrum
- Eigenvalues
- Hamiltonian systems
- Schrödinger equation
- Stability

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*145*(1), 363-377. https://doi.org/10.1090/proc/13192

**Counting spectrum via the Maslov index for one dimensional θ−periodic schrÖdinger operators.** / Jones, Christopher K.R.T.; Latushkin, Yuri; Sukhtaiev, Selim.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 145, no. 1, pp. 363-377. https://doi.org/10.1090/proc/13192

}

TY - JOUR

T1 - Counting spectrum via the Maslov index for one dimensional θ−periodic schrÖdinger operators

AU - Jones, Christopher K.R.T.

AU - Latushkin, Yuri

AU - Sukhtaiev, Selim

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We study the spectrum of the Schrödinger operators with n × n matrix valued potentials on a finite interval subject to θ−periodic boundary conditions. For two such operators, corresponding to different values of θ, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to θ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a fixed θ.

AB - We study the spectrum of the Schrödinger operators with n × n matrix valued potentials on a finite interval subject to θ−periodic boundary conditions. For two such operators, corresponding to different values of θ, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to θ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schrödinger operator for a fixed θ.

KW - Differential operators

KW - Discrete spectrum

KW - Eigenvalues

KW - Hamiltonian systems

KW - Schrödinger equation

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=84994300054&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994300054&partnerID=8YFLogxK

U2 - 10.1090/proc/13192

DO - 10.1090/proc/13192

M3 - Article

VL - 145

SP - 363

EP - 377

JO - Proceedings of the American Mathematical Society

T2 - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -