We characterize possible spectra of rank-one perturbations B a self-adjoint operator A with discrete spectrum and, in particular, prove that the may include any number real or non-real eigenvalues arbitrary algebraic multiplicity

Self-adjoint operators and their spectra play a crucial rôle in analysis and physics. For instance, in quantum physics self-adjoint operators are used to describe measurements and the spectrum represents the set of possible measurement results. Therefore, it is a natural question whether the spectrum of a self-adjoint operator can be computed from a description of the operator. We prove that gi...

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to soq(3), it acts on the q-Euclidean space that becomes a soq(3)-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on C∞ functions on R. On...

A self-adjoint operator A in a Krĕın space (K, [ · , · ]) is called partially fundamentally reducible if there exist a fundamental decomposition K = K+[+̇]K− (which does not reduce A) and densely defined symmetric operators S+ and S− in the Hilbert spaces (K+, [ · , · ]) and (K−,−[ · , · ]), respectively, such that each S+ and S− has defect numbers (1, 1) and the operator A is a self-adjoint ext...

We provide, by a resolvent Krĕın-like formula, all selfadjoint extensions of the symmetric operator S obtained by restricting the self-adjoint operator A : D(A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D(A). Neither the knowledge of S∗ nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restric...

On the half line 0; 1) we study rst order diierential operators of the form B 1 i d dx + Q(x); where B := B 1 0 0 ?B 2 ; B 1 ; B 2 2 M(n; C) are self{adjoint positive deenite matrices and Q : R + ! M(2n; C); R + := 0; 1); is a continuous self{adjoint oo{diagonal matrix function. We determine the self{adjoint boundary conditions for these operators. We prove that for each such boundary value pro...

On the half line 0; 1) we study rst order diierential operators of the form B 1 i d dx + Q(x); where B := B 1 0 0 ?B 2 ; B 1 ; B 2 2 M(n; C) are self{adjoint positive deenite matrices and Q : R + ! M(2n; C); R + := 0; 1); is a continuous self{adjoint oo{diagonal matrix function. We determine the self{adjoint boundary conditions for these operators. We prove that for each such boundary value pro...

This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we w...