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It is shown that if $X$ is a unitary operator so that a singular subspace of~$U$ is unitarily equivalent to a singular subspace of~$UX$ (or $XU$), for each unitary operator~$U$, then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman-Krein Theorem that includes the preservation of a singular spectral subspace in this context.

We obtain slow dynamics for self-adjoint semigroups and unitary evolution groups. For semigroups, the slow dynamics is for orbits, and for the average return probability in the case of unitary evolution groups. We present an application to the quantum dynamics of purely absolutely continuous systems, Comment: This is a new work that was derived from some ideas presented in the preprint "Generic properties of purely {\alpha}-Hausdorff continuous operators"

Exact Hausdorff dimensions are computed for singular continuous components of the spectral measures of a class of Schr\"odinger operators in bounded intervals.

A quantum graph model for a single sheet of graphene is extended to bilayer and trilayer Bernal-stacked graphene; the spectra are characterized and the dispersion relations explicitly obtained; Dirac cones are then proven to be present only for trilayer graphene, although the bilayer has a gapless parabolic band component. Our model rigorously exhibits basic facts from tight-binding calculations, effective two-dimensional models and a $\pi$-orbital continuum model with nearest-neighbour tunneling that have been discussed in the physics literature., Comment: J. Phys. A: Math. Theor., 29 pages

We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schr\"odinger operator, in a quasi-convex domain~$\Omega$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations., Comment: 34 pages. To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

It is shown that there exits a lower bound $\alpha>0$ to the Hausdorff dimension of the spectral measures of the one-dimensional period doubling substitution Schr\"odinger operator, and, generically in the hull of such sequence, $\alpha$ is also a lower bound to the upper packing dimension of spectral measures.

We study generic fractal properties of bounded self-adjoint operators through lower and upper generalized fractal dimensions of their spectral measures. Two groups of results are presented. Firstly, it is shown that the set of vectors whose associated spectral measures have lower (upper) generalized fractal dimension equal to zero (one) for every $q>1$ ($0

For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneously zero upper-Hausdorff and one lower-packing dimensions contains a dense $G_\delta$ subset. Applications include sets of limit-periodic operators.

It is shown that, Baire generically, the bound states of the Hamiltonian of the Hydrogen atom have spectral measures with exact $0$-lower and $1/3$-upper generalized fractal dimensions; the relation to (a weak form of) dynamical delocalization along orthonormal bases is also discussed. Such result is a consequence of the distribution of the Hamiltonian eigenvalues.

We propose a simple situation in which the magnetic Aharonov-Bohm potential influences the values of the deficiency indices of the initial Schr\"odinger operator, so determining whether the particle interacts with the solenoid or not. Even with the particle excluded from the magnetic field, the number of self-adjoint extensions of the initial Hamiltonian depends on the magnetic flux. This is a new point of view of the Aharonov-Bohm effect.

Prediction capability is considered an important issue in response surface methodology. Following the line of argument that a design should have several desirable properties we have extended an existing compound design criterion to include prediction properties. Prediction of responses and of differences in response are considered. Point and interval predictions are allowed for. Extensions of existing graphical tools for inspecting prediction performances of the designs in the whole region of experimentation are also introduced. The methods are illustrated with two examples.

Stimulated by the category theorems of Eisner and Ser\'eny in the setting of unitary and isometric $C_0$-semigroups on separable Hilbert spaces, we prove category theorems for Schr\"odinger semigroups. Specifically, we show that, to a given class of Schr\"{o}dinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the corresponding Schr\"{o}dinger operators., Comment: 12 pages, minor corrections. To appear in Zeitschrift f\"ur analysis und ihre anwendungen

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schr\"odinger operators., Comment: 17 pages, to appear in Letters in Mathematical Physics. Major changes following suggestions of the referees

We study relations between the decaying rates of operator semigroups on Hilbert spaces and some spectral properties of their respective generators; in particular, we show that the decaying rates of orbits of semigroups which are stable but not exponentially stable, typically in Baire's sense, depend on sequences of time going to infinity., Comment: 15 pages, to appear in Proceedings of the American Mathematical Society. Improved exposition. Major changes following suggestions of the referee

We add a confining potential to the Aharonov-Bohm model resulting in no contact of the particle with the solenoid (border); this is characterized by a unique self-adjoint extension of the initial Hamiltonian operator. It is shown that the spectrum of such extension is discrete and the first eigenvalue is found to be a nonconstant 1-periodic function of the magnetic flux circulation with a minimum at integers and maximum at half-integer circulations. This is a rigorous verification of the effect., Comment: It is a report of results. Proofs appear elsewhere