non-Fredholm Operators Research Articles

Solvability in the sense of sequences for some linear and nonlinear Fredholm operators with the logarithmic Laplacian

We study the solvability of certain linear and nonlinear nonhomogeneous equations in one dimension involving the logarithmic Laplacian and the transport term. In the linear case we show that the convergence in L 2 ( R ) of their right sides yields the existence and the convergence in L 2 ( R ) of the solutions. We generalize the results obtained in the earlier article of Efendiev and Vougalter [Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian. Monatsh Math. 2023] in the non-Fredholm case without the drift. In the nonlinear part of the work we demonstrate that, under the reasonable technical assumptions, the convergence in L 1 ( R ) of the integral kernels implies the existence and the convergence in L 2 ( R ) of the solutions.

Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian

Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian

The Witten index and the spectral shift function

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.

Linear and nonlinear non-Fredholm operators and their applications

<abstract><p>In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.</p></abstract>

Solvability in the sense of sequences for some fourth order non-Fredholm operators

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L2(Rd) of their right sides implies the existence and the convergence in H4(Rd) of the solutions. The problems contain the squares of the sums of second order non-Fredholm differential operators and we use the methods of the spectral and scattering theory for Schrödinger type operators. We especially emphasize that here we deal with the fourth order operators in contrast to the second order operators in [29] and investigate the dependence of the solvability conditions on the dimension of our problem when the constant a=0. We also consider the case of solvability with a single potential in an arbitrary dimension.

On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.

On Solvability in the Sense of Sequences for some Non-Fredholm Operators in Higher Dimensions

We study the solvability of some linear inhomogeneous elliptic equations and establish that, under reasonable technical conditions, the convergence in L2(ℝd) of their righthand sides yields the existence and convergence of the solutions in L2(ℝd).

Trace formulas for a class of non-Fredholm operators: A review

Take a one-parameter family of self-adjoint Fredholm operators [Formula: see text] on a Hilbert space [Formula: see text], joining endpoints [Formula: see text]. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator [Formula: see text] acting in [Formula: see text], where [Formula: see text] denotes the multiplication operator [Formula: see text] for [Formula: see text]. Most results are about the case where the operators [Formula: see text] have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators [Formula: see text], [Formula: see text], an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on [Formula: see text]: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text]. Associated to the pairs [Formula: see text] and [Formula: see text] are Krein spectral shift functions [Formula: see text] and [Formula: see text], respectively. From the trace formula, it was shown that there is a second, Pushnitski-type, formula: [Formula: see text] This can be employed to establish the desired equality, [Formula: see text] This equality was generalized to non-Fredholm operators in [14] in the form [Formula: see text] replacing the Fredholm index on the left-hand side by the Witten index of [Formula: see text] and [Formula: see text] on the right-hand side by an appropriate arithmetic mean (assuming [Formula: see text] is a right and left Lebesgue point for [Formula: see text] denoted by [Formula: see text] and [Formula: see text], respectively). But this applies only under the restrictive assumption that the endpoint [Formula: see text] is a relatively trace class perturbation of [Formula: see text] (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a [Formula: see text]-dimensional setup, where [Formula: see text] are non-Fredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitski-type formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form [Formula: see text] for the [Formula: see text]-dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain non-Fredholm situations and also point the way to the study of higher-dimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.

On a spectral flow formula for the homological index

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)|t∈R} that converges in norm to asymptotes A± at ±∞. Then under certain conditions [22] that include the assumption that the operators {D(t)=D+A(t),t∈R} all have discrete spectrum the spectral flow along the path {D(t)} can be shown to be equal to the index of ∂t+D(t) when the latter is an unbounded Fredholm operator on L2(R,H). In [16] an investigation of the index = spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In [11] the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index.In this paper we take the investigation begun in [11] much further. We show how to generalize a formula known from the setting of the L2 index theorem to the non-Fredholm setting. Restricting back to the case of selfadjoint Fredholm operators our formula extends the result of [22] in the sense of relaxing the discrete spectrum condition. It also generalizes some other Fredholm operator results of [21,16,11] that permit essential spectrum for the operators in the path. Our result may also apply however when the operators {D(t)} have essential spectrum equal to the whole real line.Our main theorem gives a trace formula relating the homological index of [7] to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalization of spectral flow to the non-Fredholm setting.

Anomalies of Dirac Type Operators on Euclidean Space

We develop by example a type of index theory for non-Fredholm operators. A general framework using cyclic homology for this notion of index was introduced in a separate article (Carev and Kaad, Topological invariance of the homological index. arXiv:1402.0475 [math.KT], 2014) where it may be seen to generalise earlier ideas of Carey–Pincus and Gesztesy–Simon on this problem. Motivated by an example in two dimensions in Bollé et al. (J Math Phys 28:1512–1525, 1987) we introduce in this paper a class of examples of Dirac type operators on \({\mathbb{R}^{2n}}\) that provide non-trivial examples of our homological approach. Our examples may be seen as extending old ideas about the notion of anomaly introduced by physicists to handle topological terms in quantum action principles, with an important difference, namely, we are dealing with purely geometric data that can be seen to arise from the continuous spectrum of our Dirac type operators.

On the solvability in the sense of sequences for some non-Fredholm operators

We investigate solvability of certain linear nonhomogeneous elliptic equa- tions and establish that under reasonable technical conditions the convergence in L 2 (R d ) of their right sides yields the existence and the convergence in H 2 (R d ) of the solutions. The problems involve the sums of second order differential operators without Fredholm property and we apply the methods of spectral and scattering theory for Schr¨ odinger type

Solvability conditions for some non-Fredholm operators

AbstractWe obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu+V(x)u–au=fwhere a ≥ 0. The conditions are formulated in terms of orthogonality of the functionfto the solutions of the homogeneous adjoint equation.

Similarity invariant semigroups generated by non-Fredholm operators

Let A ∈ B(H) be a bounded non-compact operator that is not semi-Fredholm. The similarity invariant semigroup generated by A is shown to consist of all operators that are not semi-Fredholm and satisfy obvious inequalities for the nullity and co-nullity.

Solvability conditions for elliptic problems with non-Fredholm operators

The paper is devoted to solvability conditions for linear elliptic problems with non-Fredholm operators. We show that the operator becomes normally solvable with a finite-dimensional kernel on properly chosen subspaces. In the particular case of a scalar

D-Brane Bound States Redux

We study the existence of D-brane bound states at threshold in Type II string theories. In a number of situations, we can reduce the question of existence to quadrature, and the study of a particular limit of the propagator for the system of D-branes. This involves a derivation of an index theorem for a family of non-Fredholm operators. In support of the conjectured relation between compactified eleven-dimensional supergravity and Type IIA string theory, we show that a bound state exists for two coincident zero-branes. This result also provides support for the conjectured description of M-theory as a matrix model. In addition, we provide further evidence that there are no BPS bound states for two and three-branes twice wrapped on Calabi–Yau vanishing cycles.