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 study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎The equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎Such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

In this work, we extend the analysis of problem switching controls proposed in [E. Zuazua, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 85--117]. The asks following question: Assuming that one can control a system using two or more actuators, does there exist strategy such at all times, only actuator is active? We answer positively when controlled corresponds to an analytic semigroup spanned by po...

We investigate the self-adjointness of two-dimensional Dirac operator $D$, with $quantum$-$dot$ and $Lorentz$-$scalar$ $\\delta$-$shell$ boundary conditions, on piecewise $C^2$ domains (with finitely many corners). For both models, we prove existence a unique self-adjoint realization whose domain is included in Sobolev space $H^{1/2}$, formal form free operator. The main part our paper consists...

We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for cla...

A free non-relativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Non-local bo...

We consider the problem of variation of spectral subspaces for linear self-adjoint operators under off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections.

The geometric multiplicity of each eigenvalue of a self-adjoint Sturm-Liouville problem is equal to its algebraic multiplicity. This is true for regular problems and for singular problems with limit-circle endpoints, including the case when the leading coefficient changes sign.

We prove Hölder-continuous dependence results for the difference between certain ill-posed and well-posed evolution problems in a Hilbert space. Specifically, given a positive self-adjoint operator D in a Hilbert space, we consider the ill-posed evolution problem du(t) dt = A(t,D)u(t) 0 ≤ t < T

In this short paper, the usage of truncation method to get information about essential spectrum of bounded as well as semi-bounded linear operators on separable Hilbert spaces, is investigated. In addition to this, the problem of predicting the gaps in the essential spectrum of self-adjoint operators, linear algebraically is also considered.