Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the self{adjoint extensions of H can have inside the gap J and provide methods how to construct self{adjoint extensions of H with prescribed spectral properties inside J . Under some weak assumptions about the operator H which are satised...

Abstract The paper suggests a new technique for solution of the inverse problem for Maxwell’s equations in a dissipating medium. Being non-self-adjoint, the problem generally requires computing an adjoint operator on each step of gradient descent minimization procedure. The suggested approach introduces a conjugation operator and develops a representation of the system of Maxwell’s equations ba...

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...

We define a class of pseudo-ergodic non-self-adjoint Schrödinger operators acting in spaces l 2 (X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson model acting on l 2 (Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for su...

The norm convergence of the Trotter{Kato product formula is established with ultimate optimal error bound for the self-adjoint semigroup generated by the operator sum of two self-adjoint operators. A generalization is also given to the operator sum of several self-adjoint operators.

We continue here the analysis of the previous paper of the Wheeler-DeWitt constraint operator for four-dimensional, Lorentzian, non-perturbative, canonical vacuum quantum gravity in the continuum. In this paper we derive the complete kernel, as well as a physical inner product on it, for a non-symmetric version of the Wheeler-DeWitt operator. We then define a symmetric version of the Wheeler-De...

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q≡Adn(x;m), where m∈(0,1) and A can be taken to positive without loss generality, for a non-self-adjoint Dirac operator L, which connects two well-known limiting cases the plane wave (m=0) sech potential (m=1). show that, if A∈N, then spectrum consists R plus 2A Schwarz symmetric segments (bands) on i...