Essential Self-Adjointness of Anticommutative Operators

A new characterization of the positive self-adjoint extensions of symmetric operators, T 0 T_0 , is presented, which is based on the Friedrichs extension of T 0 , T_0, a direct sum decomposition of domain of the adjoint T 0 ∗ T_0^{*} and the boundary mapping of T 0 ∗ T_0^{*} . In applying this result to ordinary differential equations, we characterize all positive self-adjoint extensions of symmetric regular differential operators of order 2 n 2n in terms of boundary conditions.

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian.

We consider a point interaction model for two particles in an external field, which is similar to the Skornyakov-Ter-Martirosyan model and is treated using the theory of self-adjoint extensions of symmetric operators. The corresponding energy operator has an infinite set of bound states with the energy values receding to-∞.

Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state. In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.

Application of zero-radius potentials to problems of diffraction by small inhomogeneities in elastic plates

A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian (J-self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨n, n⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J-self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.

Spectral analysis of dissipative Dirac operators with general boundary conditions

In this thesis we introduce the S-theory. We apply the S-theory to variational problems of immersed Riemann surfaces, Elasticity theory and Spin theory. 
\nFrom classical mechanics it is known that geometric problems can be substantially simplified by symmetry considerations. The Noether theorem describes in this case how to find conservation laws from the invariance property of the observed energy. At first, the motivation was to give the space of immersions M = {f : M → R3}, for an oriented Riemannian manifold M of dimension less than or equal to three, an Euclidean invariant description. For geometric energies, i.e. energies which are invariant under Euclidean transformations, one would be able to derive conservation laws for the respective variational problems. It is a well known fact that immersed space curves are uniquely determined by their curvature functions (κ1,κ2,τ) up to Euclidean transformations. In the case of immersed surfaces and immersed 3-manifolds the situation becomes more complicated and leads to a non conformal deformation theory (S-Theorie). 
\nThe S-Theory builds the foundations of this thesis. Starting from a Riemannian manifold M and a reference metric ⟨ , ⟩ one can model any other Riemmanian metric g through a positive definite and self adjoint operator S via g = ⟨S,S⟩. 
\nThe operator S is an isometry, i.e. g = S∗⟨,⟩. From this point of view we compute the Levi- Civita connection with respect to g out of Levi-Civita connection of the reference metric and the operator S. 
\nFurther we introduce Spin bundles over two and three dimensional oriented Riemannian manifolds. Thereby Spin bundles, in contrast to most of the literature, have additionally a quaternionic structure. Many formulas become clearer and more accessible. We show that any Spin bundle Σ over (M, ⟨ , ⟩) has a unique Spin connection and compute via the S-Theory the deformed Spin connection with respect to the metric g. Eventually we consider the induced Spin bundle of an immersion f : M → R3 and compute the corresponding Spin connection and find a new interpretation of the Gauss-Codazzi equation. We introduce the Dirac operator and compute its deformation with respect to the metric g. 
\nWe are now able to give the space of immersed surfaces M = {f : M → R3} the desired geometric description. We compute the normal space of M and formulate the Noether theorem. As an application we compute conservation laws for the Willmore functional. 
\nFinally we introduce the intrinsic version of the famous distance-squared energy on a n-dimensional Riemannian manifold. We introduce the corresponding stress tensor and show that it’s closeness is a characterization for critical points of the energy. Thereby the S-theory provides a fundamental concept for the understanding of the stress tensor. In the case of a Riemann surface, the S-theory provides a canonical harmonic rotation 1-form for critical points of the distance-squared energy.

Kaufman’s representation theorem is that a closed operator S with a dense domain in Hilbert space H is represented by a quotient S = B/(1 − B * B)1/2 for a unique contraction B. When S is a symmetric operator, what is a condition of the spectrum of B to admit selfadjoint extensions of S? In this note, it is shown that if there are no negative real points in the spectrum of B, then S has selfadjoint extensions.

We formulate a theorem saying that the Dirac operator corresponding to a one-electron atomic ion is essentially self-adjoint on the usual domain, provided that the nuclear charge Z is less than 118. Furthermore, for such nuclear charges the domains of the closure of the free particle and total Dirac operators are equal. In the present part I of this paper we prove this theorem for the part of the operator over each of the usual reducing subspaces.

It is shown by concrete examples that in the case of Hermitian (not densely defined) operators their self-adjoint bound-preserving extensions exist not always. A sufficient condition for the existence of such extensions is established.

Generalized uncertainty principle and self-adjoint operators

An analytic characterization of the eigenvalues of self-adjoint extensions

Let S be a symmetric operator with finite and equal defect numbers d in the Hilbert space \(\mathfrak{H}\), and with a boundary triplet \((\mathbb{C}^d, \Gamma_1, \Gamma_2)\). Following the method of E.A. Coddington, we describe all self-adjoint extensions \(\tilde{A}\) of S in a Hilbert space \(\tilde{\mathfrak{H}}\;=\;\mathfrak{H}\oplus \mathfrak{H}_1\) where \(\mathrm{dim}\;\mathfrak{H}_1\;<\;\infty\). The parameters in this description are matrices \(\mathcal{A,B,U,V}\;\mathrm{and}\;\mathcal{E}\), where \(\mathcal{A}\;\mathrm{and}\;\mathcal{B}\) determine the compression \(A_0\;\mathrm{of}\;\tilde{A}\;\mathrm{to}\;\mathfrak{H}\). According to a result of W. Stenger, this compression \(A_0\) is self-adjoint. Being a canonical self-adjoint extension of S, \(A_0\) can be chosen as the fixed extension in M.G. Krein’s formula for the description of all generalized resolvents of S. Among other results, we describe those parameters which in Krein’s formula correspond to self-adjoint extensions of S having \(A_0\) as their compression to \(\mathfrak{H}\).