Let H be a Hilbert space. For a bounded operator A : H → H its Hilbert space adjoint is an operator A∗ : H → H such that 〈Ax, y〉 = 〈x,A∗y〉 for all x, y ∈ H. We say that A is bounded self adjoint if A = A∗. In this chapter we discussed several results about the spectrum of a bounded self adjoint operator on a Hilbert space. We emphasize that in this chapter A is bounded, there is also a notion o...

One-dimensional δ ′ -function potential is discussed in the framework of Green’s function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green’s function for this case is crucially dependent on the boundary conditions which are provided by self-adjoint extension method. The most general Green’s function which contains four real self-adjoint extension pa...

For any even positive integer 2n and any positive integermwe construct a class of regular self-adjoint and non-self-adjoint boundary value problems whose spectrum consists of at most (2n − 1)m + 1 eigenvalues. Our main result reduces to previously known results for the cases n = 1 and n = 2. In the self-adjoint case with separated boundary conditions this upper bound can be improved to n(m + 1)...

Abstract We develop a categorical index calculus for elliptic symbol families. The categorified problems we consider are secondary version of the traditional problem expressing class in K-theory terms differential-topological data. They include orientation moduli spaces as well similar skew-adjoint and self-adjoint operators. main result this paper is an excision principle that allows compariso...

This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum ...

Given the symmetric operator AN obtained by restricting the self-adjoint operator A to N , a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A N and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of AN and their resolvents.