We discuss the discrete spectrum induced by bulges on threadlike mesoscopic objects, using two models, a continuous hard-wall waveguide and a discrete tight-binding model with two sorts of atomic orbitals. We show that elongated bulges induce numerous quasibound states. In the discrete model we also evaluate the probability of transition between the localized states and extended ones of the valence band. We suggest this as a mechanism governing the porous-silicon luminescence. In addition, the model reproduces the dominance of nonradiative transitions, blueshift for finer textures, and luminiscence suppression at low temperatures.

Let <i>A</i> and <i>B</i> be non-negative self-adjoint operators in a separable Hilbert space such that their form sum <i>C</i> is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the <i>L</i> ²-norm, that is, one has <div class="gsh_dspfr"> \lim_{no+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} ight)^nh - e^{-itC}hight\|^2dt = 0

We analyse two-dimensional Schrdinger operators with the potential| xy| p (x 2+ y 2) p/(p+ 2) where p 1 and 0. We show that there is a critical value of such that the spectrum for &lt; crit is bounded below and purely discrete, while for &gt; crit it is unbounded from below. In the subcritical case, we prove upper and lower bounds for the eigenvalue sums.

We investigate approximations of the vertex coupling on a star-shaped graph by families of operators with singularly scaled rank-one interactions. We find a family of vertex couplings, generalizing the <i></i>-interaction on the line, and show that with a suitable choice of the parameters they can be approximated in this way in the norm-resolvent sense. We also analyze spectral properties of the involved operators and demonstrate the convergence of the corresponding on-shell scattering matrices.

We consider a two-dimensional particle of charge e interacting with a homogeneous magnetic field perpendicular to the plane and a potential well which is transported along a closed loop in the plane. We show that a bound state corresponding to a simple isolated eigenvalue acquires at that Berry's phase equal to 2 sgn e times the number of flux quanta through the oriented area encircled by the loop. We also argue that this is a purely quantum effect since the corresponding Hannay angle is zero.