A new characterization of the positive self-adjoint extensions of symmetric operators, T0, is presented, which is based on the Friedrichs extension of T0, a direct sum decomposition of domain of the adjoint T ∗ 0 and the boundary mapping of T ∗ 0 . In applying this result to ordinary differential equations, we characterize all positive self-adjoint extensions of symmetric regular differential o...

in this paper, we consider weighted composition operators betweenmeasurable differential forms and then some classic properties of these operators are characterized.

We obtain bounds on the complex eigenvalues of nonself-adjoint Schrödinger operators with complex potentials, and also on the complex resonances of self-adjoint Schrödinger operators. Our bounds are compared with numerical results, and are seen to provide useful information.

We review somemore and less recent results concerning bounds on nonlinear eigenvalues NLEV for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV as the norm of the eigenvector tends to zero in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-S...

In the gap topology, unbounded self-adjoint Fredholm operators on a Hilbert space have third homotopy group integers. We realise generator explicitly, using family of Dirac half-line, which arises naturally in Weyl semimetals solid-state physics. A “Fermi gerbe” geometrically encodes how discrete spectral data interpolate between essential gaps. Its non-vanishing Dixmier–Douady invariant protec...

On the half line [0,∞) we study first order differential operators of the form B 1 i d dx +Q(x), where B := ( B1 0 0 −B2 ) , B1, B2 ∈ M(n,C) are self–adjoint positive definite matrices and Q : R+ → M(2n,C), R+ := [0,∞), is a continuous self–adjoint off–diagonal matrix function. We determine the self–adjoint boundary conditions for these operators. We prove that for each such boundary value prob...

On the half line [0,∞) we study first order differential operators of the form B 1 i d dx +Q(x), where B := ( B1 0 0 −B2 ) , B1, B2 ∈ M(n,C) are self–adjoint positive definite matrices and Q : R+ → M(2n,C), R+ := [0,∞), is a continuous self–adjoint off–diagonal matrix function. We determine the self–adjoint boundary conditions for these operators. We prove that for each such boundary value prob...