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.

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1994/1/1

Y1 - 1994/1/1

N2 - We consider uniformly elliptic divergence type operators A=−Σ∂jaij(x)∂i with bounded, Lipschitz continuous coefficients, acting in the Hilbert space L2(Rν). 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 the family of operators A+λB=−Σ∂j(aij(x)+λbij(x))∂i, λ≥0, where A is supposed to have a spectral gap, while (bij)≥0 and bij(x)→0, as x→∞. One of our main results assures that discrete eigenvalues of A+λB move into the gap, as λ increases, if the support of the matrix function (bij) is large enough. 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 Schrödinger case.

AB - We consider uniformly elliptic divergence type operators A=−Σ∂jaij(x)∂i with bounded, Lipschitz continuous coefficients, acting in the Hilbert space L2(Rν). 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 the family of operators A+λB=−Σ∂j(aij(x)+λbij(x))∂i, λ≥0, where A is supposed to have a spectral gap, while (bij)≥0 and bij(x)→0, as x→∞. One of our main results assures that discrete eigenvalues of A+λB move into the gap, as λ increases, if the support of the matrix function (bij) is large enough. 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 Schrödinger 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

U2 - 10.3233/ASY-1994-8401

DO - 10.3233/ASY-1994-8401

M3 - Article

AN - SCOPUS:0028444852

SN - 0921-7134

VL - 8

SP - 311

EP - 344

JO - Asymptotic Analysis

JF - Asymptotic Analysis

IS - 4

ER -