A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

Given a self-adjoint operator A:D(A)\subseteq\calH\to\calH and a continuous linear operator \tau:D(A)\to\X with Range \tau'\cap\calH' ={0}, \X a Banach space, we explicitly construct a family $A^\tau_\Theta$ of self-adjoint operators such that any $A^\tau_\Theta$ coincides with the original $A$ on the kernel of $\tau$. Such a family is obtained by giving a Kre\u\i n-like formula where the role of the deficiency spaces is played by the dual pair (\X,\X'); the parameter $\Theta$ belongs to the space of symmetric operators from \X' to \X. When \X=\C one recovers the ``\calH_{-2} -construction'' of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which \calH=L^2(\RE^n) and $\tau$ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.Comment: Proposition 2.1 revised. Remarks 2.15 and 2.16 added. 38 pages. To appear in Journal of Functional Analysi

Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Given the symmetric operator $A_N$ 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 $A_N$ and their resolvents.Comment: Misprints corrected. To appear in Methods of Functional Analysis and Topolog

Nonlinear Maximal Monotone Extensions of Symmetric Operators

Given a linear semi-bounded symmetric operator $S\ge -\omega$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_\Theta$ of type $\lambda>\omega$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $\lambda$) which coincide with the Friedrichs extension of $S$ on a convex set containing ${\mathscr D}(S)$. The extension parameter $\Theta\subset{\mathfrak h}\times{\mathfrak h}$ ranges over the set of nonlinear maximal monotone relations on an auxiliary Hilbert space $\mathfrak h$ isomorphic to the deficiency subspace of $S$. Moreover $A_\Theta+\lambda$ is a sub-potential operator (i.e. is the sub-differential of a lower semicontinuos convex function) whenever $\Theta$ is sub-potential. Examples describing Laplacians with nonlinear singular perturbations supported on null sets and Laplacians with nonlinear boundary conditions on a bounded set are given.Comment: Revised final version. To appear in Journal of Evolution Equation

Singular Perturbations of Abstract Wave equations

Given, on the Hilbert space \H_0, the self-adjoint operator $B$ and the skew-adjoint operators $C_1$ and $C_2$, we consider, on the Hilbert space \H\simeq D(B)\oplus\H_0, the skew-adjoint operator W=[\begin{matrix} C_2&\uno -B^2&C_1\end{matrix}] corresponding to the abstract wave equation $\ddot\phi-(C_1+C_2)\dot\phi=-(B^2+C_1C_2)\phi$. Given then an auxiliary Hilbert space \fh and a linear map \tau:D(B^2)\to\fh with a kernel \K dense in \H_0, we explicitly construct skew-adjoint operators $W_\Theta$ on a Hilbert space \H_\Theta\simeq D(B)\oplus\H_0\oplus \fh which coincide with $W$ on \N\simeq\K\oplus D(B). The extension parameter $\Theta$ ranges over the set of positive, bounded and injective self-adjoint operators on \fh. In the case $C_1=C_2=0$ our construction allows a natural definition of negative (strongly) singular perturbations $A_\Theta$ of $A:=-B^2$ such that the diagram $$\begin{CD} W @>>> W_\Theta @AAA @VVV A@>>> A_\Theta \end{CD}$$ is commutative.Comment: Revised version. Misprints corrected. New examples and a digression on a possible application to the electrodynamics of a point particle added. Accepted for publication in Journal of Functional Analysi

Direct sums of trace maps and self-adjoint extensions

We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an alternative approach to results recently obtained by Malamud-Neidhardt and Kostenko-Malamud using regularized direct sums of boundary triplets.Comment: Final version. To appear in: S. Albeverio (ed.), Singular Perturbation Theory, Analysis, Geometry, and Stochastic, special issue of Arab. J. Math. (Springer

On the common Point Spectrum of pairs of Self-Adjoint Extensions

Given two different self-adjoint extensions of the same symmetric operator, we analyse the intersection of their point spectra. Some simple examples are provided.Comment: Two references added. To appear in Methods of Functional Analysis and Topology, special issue dedicated to Vladimir Koshmanenko on the occasion of his 70th birthda

Convergence of symmetric diffusions on Wiener spaces

We prove convergence of symmetric diffusions on Wiener spaces by using stopping times arguments and capacity techniques. The drifts of the diffusions can be singular, we require the densities of the processes to be neither bounded from above nor away from zero

Time dependent delta-prime interactions in dimension one

We solve the Cauchy problem for the Schr\"odinger equation corresponding to the family of Hamiltonians $H_{\gamma(t)}$ in $L^{2}(\mathbb{R})$ which describes a $\delta'$-interaction with time-dependent strength $1/\gamma(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapsto\gamma(t)$ belongs to the fractional Sobolev space $H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.Comment: minor changes, 10 page