We study bound states of the two-dimensional Helmholtz equations with Dirichlet boundary conditions in an open geometry given by two straight leads of the same width which cross at an angle . Such a four-terminal junction with a tunable can realized experimentally if a right-angle structure is filled by a ferrite. It is known that for = 90 there is one proper bound state and one eigenvalue embedded in the continuum. We show that the number of eigenvalues becomes larger with increasing asymmetry and the bound-state energies are increasing as functions of in the interval (0, 90). Moreover, states which are sufficiently strongly bound exist in pairs with a small energy difference and opposite parities. Finally, we discuss how the bound states transform with increasing into quasibound states with a complex wave vector.

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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.

The human cerebellum has recently been discovered to contribute to cognition and emotion beyond the planning and execution of movement, suggesting its functional heterogeneity. We aimed to identify the functional parcellation of the cerebellum using information from resting-state functional magnetic resonance imaging (rs-fMRI). For this, we introduced a new data-driven decomposition-based functional parcellation algorithm, called Sparse Dictionary Learning Clustering (SDLC). SDLC integrates dictionary learning, sparse representation of rs-fMRI, and k-means clustering into one optimization problem. The dictionary is comprised of an over-complete set of time course signals, with which a sparse representation of rs-fMRI signals can be constructed. Cerebellar functional regions were then identified using k-means clustering based on the sparse representation of rs-fMRI signals. We solved SDLC using a multi-block hybrid proximal alternating method that guarantees strong convergence. We evaluated the reliability of SDLC and benchmarked its classification accuracy against other clustering techniques using simulated data. We then demonstrated that SDLC can identify biologically reasonable functional regions of the cerebellum as estimated by their cerebello-cortical functional connectivity. We further provided new insights into the cerebello-cortical functional organization in children.

We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Kreins formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.