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
Y1 - 1994
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.
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U2 - 10.3233/ASY-1994-8401
DO - 10.3233/ASY-1994-8401
M3 - Article
AN - SCOPUS:0028444852
VL - 8
SP - 311
EP - 344
JO - Asymptotic Analysis
JF - Asymptotic Analysis
SN - 0921-7134
IS - 4
ER -