The index formula and the spectral shift function for relatively trace class perturbations

• Published in: Advances in mathematics (New York. 1965) Vol. 227; no. 1; pp. 319 - 420
• Main Authors: Gesztesy, Fritz, Latushkin, Yuri, Makarov, Konstantin A., Sukochev, Fedor, Tomilov, Yuri
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-05-2011
• Subjects: Fredholm index; Perturbation determinants; Relative trace class perturbations; Spectral flow; Spectral shift function; secondary; Perturbation determinants; Spectral shift function; Relative trace class perturbations; Fredholm index; Spectral flow; primary
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2011.01.022

We compute the Fredholm index,
index
(
D
A
)
, of the operator
D
A
=
(
d
/
d
t
)
+
A
on
L
2
(
R
;
H
)
associated with the operator path
{
A
(
t
)
}
t
=
−
∞
∞
, where
(
A
f
)
(
t
)
=
A
(
t
)
f
(
t
)
for a.e.
t
∈
R
, and appropriate
f
∈
L
2
(
R
;
H
)
, via the spectral shift function
ξ
(
⋅
;
A
+
,
A
−
)
associated with the pair
(
A
+
,
A
−
)
of asymptotic operators
A
±
=
A
(
±
∞
)
on the separable complex Hilbert space
H
in the case when
A
(
t
)
is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator
A
−
.
We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function
ξ
(
⋅
;
A
+
,
A
−
)
for the pair
(
A
+
,
A
−
)
, and the corresponding spectral shift function
ξ
(
⋅
;
H
2
,
H
1
)
for the pair of operators
(
H
2
,
H
1
)
=
(
D
A
D
A
⁎
,
D
A
⁎
D
A
)
in this relative trace class context,
ξ
(
λ
;
H
2
,
H
1
)
=
1
π
∫
−
λ
1
/
2
λ
1
/
2
ξ
(
ν
;
A
+
,
A
−
)
d
ν
(
λ
−
ν
2
)
1
/
2
for a.e.
λ
>
0
.
This formula is then used to identify the Fredholm index of
D
A
with
ξ
(
0
;
A
+
,
A
−
)
. In addition, we prove that
index
(
D
A
)
coincides with the spectral flow
SpFlow
(
{
A
(
t
)
}
t
=
−
∞
∞
)
of the family
{
A
(
t
)
}
t
∈
R
and also relate it to the (Fredholm) perturbation determinant for the pair
(
A
+
,
A
−
)
:
index
(
D
A
)
=
SpFlow
(
{
A
(
t
)
}
t
=
−
∞
∞
)
=
ξ
(
0
;
A
+
,
A
−
)
=
π
−
1
lim
ε
↓
0
Im
(
ln
(
det
H
(
(
A
+
−
i
ε
I
)
(
A
−
−
i
ε
I
)
−
1
)
)
)
=
ξ
(
0
+
;
H
2
,
H
1
)
,
with the choice of the branch of
ln
(
det
H
(
⋅
)
)
on
C
+
such that
lim
Im
(
z
)
→
+
∞
ln
(
det
H
(
(
A
+
−
z
I
)
(
A
−
−
z
I
)
−
1
)
)
=
0
.
We also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.