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• W2227223559 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 $t to pm infty$. 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." @default.
• W2227223559 created "2016-06-24" @default.
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• W2227223559 date "2015-09-04" @default.
• W2227223559 modified "2023-09-27" @default.
• W2227223559 title "On the Index of a Non-Fredholm Model Operator" @default.
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