### Abstract

We consider uniformly elliptic divergence type operators A = -SUM ∂_{j}a_{ij}(x)∂_{i} with bounded, Lipschitz continuous coefficients, acting in a Hilbert space L_{2}(R^{v}). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study a family of operators. In addition, we analyze the connection between decay properties of the coefficient matrix (b_{ij}) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrodinger case.

Original language | English (US) |
---|---|

Pages (from-to) | 311-344 |

Number of pages | 34 |

Journal | Asymptotic Analysis |

Volume | 8 |

Issue number | 4 |

State | Published - Jun 1994 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Asymptotic Analysis*,

*8*(4), 311-344.

**On the existence of eigenvalues of a divergence-form operator A+ λB in a gap of σ(A).** / Alama, S.; Avellaneda, M.; Deift, P. A.; Hempel, R.

Research output: Contribution to journal › Article

*Asymptotic Analysis*, vol. 8, no. 4, pp. 311-344.

}

TY - JOUR

T1 - On the existence of eigenvalues of a divergence-form operator A+ λB in a gap of σ(A)

AU - Alama, S.

AU - Avellaneda, M.

AU - Deift, P. A.

AU - Hempel, R.

PY - 1994/6

Y1 - 1994/6

N2 - We consider uniformly elliptic divergence type operators A = -SUM ∂jaij(x)∂i with bounded, Lipschitz continuous coefficients, acting in a Hilbert space L2(Rv). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study a family of operators. In addition, we analyze the connection between decay properties of the coefficient matrix (bij) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrodinger case.

AB - We consider uniformly elliptic divergence type operators A = -SUM ∂jaij(x)∂i with bounded, Lipschitz continuous coefficients, acting in a Hilbert space L2(Rv). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study a family of operators. In addition, we analyze the connection between decay properties of the coefficient matrix (bij) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrodinger case.

UR - http://www.scopus.com/inward/record.url?scp=0028444852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028444852&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0028444852

VL - 8

SP - 311

EP - 344

JO - Asymptotic Analysis

JF - Asymptotic Analysis

SN - 0921-7134

IS - 4

ER -