TY - JOUR
T1 - On the Witten index in terms of spectral shift functions
AU - Carey, Alan
AU - Gesztesy, Fritz
AU - Potapov, Denis
AU - Sukochev, Fedor
AU - Tomilov, Yuri
N1 - Publisher Copyright:
© 2017, Hebrew University Magnes Press.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - We study the model operator DA = (d/dt) + A in L2(R;H) associated with the operator path A(t)t=−∞ ∞, where (Af)(t) = A(t)f(t) for a.e. t ∈ R, and appropriate f ∈ L2(R;H) (with H a separable, complex Hilbert space). Denoting by A± the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A-, and no discrete spectrum assumptions are made on A±. Introducing H1 = DA*DA, H2 = DADA*, the resolvent and semigroup regularized Witten indices of DA, denoted by Wr(DA) and Ws(DA), are defined by (Formula Presented.), whenever these limits exist. These regularized indices coincide with the Fredholm index of DA whenever the latter is Fredholm. In situations where DA ceases to be a Fredholm operator in L2(R;H) we compute its resolvent (resp., semigroup) regularizedWitten index in terms of the spectral shift function ξ(•; A+, A-) associated with the pair (A+, A-) as follows: Assuming 0 to be a right and a left Lebesgue point of ξ(•; A+, A-), denoted by ξL(0+; A+, A-) and ξL(0-; A+, A-), we prove that 0 is also a right Lebesgue point of ξ(•; H2, H1), denoted by ξL(0+; H2, H1), and that (Formula Presented.), the principal result of this paper. In the special case where dim(H) < ∞, we prove that the Witten indices of DA are either integer, or half-integer-valued.
AB - We study the model operator DA = (d/dt) + A in L2(R;H) associated with the operator path A(t)t=−∞ ∞, where (Af)(t) = A(t)f(t) for a.e. t ∈ R, and appropriate f ∈ L2(R;H) (with H a separable, complex Hilbert space). Denoting by A± the norm resolvent limits of A(t) as t → ±∞, our setup permits A(t) in H to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator A-, and no discrete spectrum assumptions are made on A±. Introducing H1 = DA*DA, H2 = DADA*, the resolvent and semigroup regularized Witten indices of DA, denoted by Wr(DA) and Ws(DA), are defined by (Formula Presented.), whenever these limits exist. These regularized indices coincide with the Fredholm index of DA whenever the latter is Fredholm. In situations where DA ceases to be a Fredholm operator in L2(R;H) we compute its resolvent (resp., semigroup) regularizedWitten index in terms of the spectral shift function ξ(•; A+, A-) associated with the pair (A+, A-) as follows: Assuming 0 to be a right and a left Lebesgue point of ξ(•; A+, A-), denoted by ξL(0+; A+, A-) and ξL(0-; A+, A-), we prove that 0 is also a right Lebesgue point of ξ(•; H2, H1), denoted by ξL(0+; H2, H1), and that (Formula Presented.), the principal result of this paper. In the special case where dim(H) < ∞, we prove that the Witten indices of DA are either integer, or half-integer-valued.
UR - http://www.scopus.com/inward/record.url?scp=85021414571&partnerID=8YFLogxK
U2 - 10.1007/s11854-017-0003-x
DO - 10.1007/s11854-017-0003-x
M3 - Article
SN - 0021-7670
VL - 132
SP - 1
EP - 61
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
IS - 1
ER -