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On the Index of a Non-Fredholm Model Operator | | Alan Carey
; Fritz Gesztesy
; Galina Levitina
; Fedor Sukochev
; | | Date: |
4 Sep 2015 | | Abstract: | Let ${A(t)}_{t in mathbb{R}}$ be a path of self-adjoint Fredholm
operators in a Hilbert space $mathcal{H}$, joining endpoints $A_pm$ as $t o
pm infty$. Computing the index of the operator $D_A= (d/d t) + A$ acting in
$L^2(mathbb{R}; mathcal{H})$, where $A = int_{mathbb{R}}^{oplus} dt ,
A(t)$, and its relation to spectral flow along this path, has a long history.
While most of the latter focuses on the case where $A(t)$ all have purely
discrete spectrum, we now particularly study situations permitting essential
spectra.
Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we consider spectral
shift functions $xi(, cdot ,; A_+, A_-)$ and $xi(, cdot , ; H_2, H_1)$
associated with the pairs $(A_+, A_-)$ and $(H_2,H_1)$. Assuming $A_+$ to be a
relatively trace class perturbation of $A_-$ and $A_{pm}$ to be Fredholm, the
value $xi(0; A_-, A_+)$ was shown in [14] to represent the spectral flow along
the path ${A(t)}_{tin mathbb{R}}$ while that of $xi(0_+; H_1,H_2)$ yields
the Fredholm index of $D_A$. The fact, proved in [14], that these values of the
two spectral functions are equal, resolves the index = spectral flow question
in this case.
When the path ${A(t)}_{t in mathbb{R}}$ consists of differential
operators, the relatively trace class perturbation assumption is violated. The
simplest assumption that applies (to differential operators in
(1+1)-dimensions) is a relatively Hilbert-Schmidt perturbation. This is not
just an incremental improvement. In fact, the approximation method we employ
here to make this extension is of interest in any dimension. Moreover we
consider $A_pm$ which are not necessarily Fredholm and we establish that the
relationships between the two spectral shift functions for the pairs $(A_+,
A_-)$ and $(H_2,H_1)$ found in all of the previous papers [9], [14], and [22]
can be proved in the non-Fredholm case. | | Source: | arXiv, 1509.1580 | | Services: | Forum | Review | PDF | Favorites | |
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