Non-compact Perturbations Research Articles

Spectral shift for relative Schatten class perturbations

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators H and V such that V is bounded and V(H-iI)^{-1} belongs to a Schatten–von Neumann ideal \mathcal{S}^n of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of V\in\mathcal{S}^n and that it is unique up to a polynomial summand of order n-1 . Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.

Non-compact perturbations of coercive functionals and applications

Non-compact perturbations of coercive functionals and applications

Non-compact perturbations of the spectrum of multipliers given with functions

Non-compact perturbations of the spectrum of multipliers given with functions

Lipschitz estimates for functions of Dirac and Schrödinger operators

We establish new Lipschitz-type bounds for functions of operators with noncompact perturbations that produce Schatten class resolvent differences. The results apply to suitable perturbations of Dirac and Schrödinger operators, including some long-range and random potentials, and to important classes of test functions. The key feature of these bounds is an explicit dependence on the Lipschitz seminorm and decay parameters of the respective scalar functions and, in the case of Dirac and Schrödinger operators, on the Lp- or ℓp(L2)-norm of the potential.

Rank-one perturbations and Anderson-type Hamiltonians

Motivated by applications of the discrete random Schr\"odinger operator, mathematical physicists and analysts, began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $$H_\omega = H + V_\omega$$ on a separable Hilbert space $\mathcal{H}$, where the perturbation is given by $$V_\omega = \sum_n \omega_n (\cdot, \varphi_n)\varphi_n$$ with a sequence $\{\varphi_n\}\subset\mathcal{H}$ and independent identically distributed random variables $\omega_n$. We show that the the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank one perturbation. This result connects one of the least trackable perturbation problem (with almost surely non-compact perturbations) with one where the perturbation is `only' of rank one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has non-zero Lebesgue measure.

Convex Sweeping Processes with Noncompact Perturbations and with Delay in Banach Spaces

We prove two results concerning the existence of solutions for functional differential inclusions that are governed by sweeping processes, with noncompact valued perturbations in Banach spaces. Indeed, we have two goals. The first is to give a technique that allows considering sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time. The second is to give a technique to overcome the arising problem from the nonlinearity of the normalized mappings, when we deal with sweeping processes with noncompact valued perturbations and associated with a multivalued function depending on time and state.

Bifurcation Results for a Class of Perturbed Fredholm Maps

AbstractWe prove a global bifurcation result for an equation of the type "Equation missing", where "Equation missing" is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset "Equation missing" of "Equation missing", "Equation missing" are "Equation missing" and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions "Equation missing" with "Equation missing", whose closure contains a trivial solution "Equation missing". The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called "Equation missing"-Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.

Survival of the black hole’s Cauchy horizon under noncompact perturbations

We study numerically the evolution of spacetime, and in particular of a spacetime singularity, inside a black hole under a class of perturbations of noncompact support. We use a very simplified toy model of a spherical charged black hole which is perturbed nonlinearly by a self-gravitating, spherical scalar field. The latter grows logarithmically with advanced time along an outgoing characteristic hypersurface. We find that for that class of perturbations a portion of the Cauchy horizon survives as a noncentral, null singularity.

On Asymptotic Behavior at Infinity and the Finite Section Method for Integral Equations on the Half-Line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

On the homotopy property of degree for multivalued noncompact maps

Over the last few years, various extensions of the topological degree of a mapping have been made so as to include non-compact perturbations of the identity. One such extension, which employs compactness conditions, has been to the class of limit compact maps which were extensively studied by Sadovsky [7]. The class is a large one as it contains all compact mappings, contraction mappings and, more generally, condensing mappings. Sadovsky [7] gives a theory of degree for maps of the form I-f, where f is limit compact, and this was extended independently and with different methods by Petryshyn and Fitzpatrick [4] and the author [9] to allow f to be a multi-valued mapping. A refinement of the methods of [9] was given by Vanderbauwhede [8].

On the homotopy property of degree for multivalued noncompact maps

Over the last few years, various extensions of the topological degree of a mapping have been made so as to include non-compact perturbations of the identity. One such extension, which employs compactness conditions, has been to the class of limit compact maps which were extensively studied by Sadovsky [7]. The class is a large one as it contains all compact mappings, contraction mappings and, more generally, condensing mappings. Sadovsky [7] gives a theory of degree for maps of the form I-f, where f is limit compact, and this was extended independently and with different methods by Petryshyn and Fitzpatrick [4] and the author [9] to allow f to be a multi-valued mapping. A refinement of the methods of [9] was given by Vanderbauwhede [8].

Existence results for equations involving noncompact perturbations of Fredholm mappings with applications to differential equations

Existence results for equations involving noncompact perturbations of Fredholm mappings with applications to differential equations