We study Lévy flights with arbitrary index 0<μ≤2 inside a potential well of infinite depth. Such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrödinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain D, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum.

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We show that in nontypical ferroelectric substances (having nonperovskite crystalline structure and hence no soft phonon mode) such as ZnO:Li, Be, Mg, the ferroelectricity might appear due to indirect interaction of dipoles, formed by Li, Be, or Mg off-center impurities, via free charge carriers. Our estimations show that the typical semiconducting concentration of the carriers like 1017 cm−3 suffices for effect to occur. We have also shown that the properties of impurity-generated ferroelectricity depend on the difference in the ionic radii of the impurity and host lattice ion as well as on their concentrations. Namely, the growing amount of Li and Be promotes ferroelectricity, while the same for Mg inhibits it. Our calculations of spontaneous polarization and ferroelectric phase transition temperature in the above nontypical ferroelectrics as the functions of concentrations of impurity dipoles and carriers capture well the main peculiarities of all available experimental data.

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Strongly correlated electron systems; heavy fermions. PACS. 74.20.Fg -BCS theory and its development. PACS. 74.25.Jb -Electronic structure.Abstract. -We demonstrate, that the main universal features of the low temperature experimental H − T phase diagram of CeCoIn5 and other heavy-fermion metals can be well explained using Landau paradigm of quasiparticles. The main point of our theory is that above quasiparticles form so-called fermion-condensate state, achieved by a fermion condensation quantum phase transition (FCQPT). When a heavy fermion liquid undergoes FCQPT, the fluctuations accompanying above quantum critical point are strongly suppressed and cannot destroy the quasiparticles. The comparison of our theoretical results with experimental data on CeCoIn5 have shown that the electronic system of above substance provides a unique opportunity to study the relationship between quasiparticles properties and non-Fermi liquid behavior.( * )

We analyze spectral properties of the ultrarelativistic (Cauchy) operator |∆| 1/2 , provided its action is constrained exclusively to the interior of the interval [−1, 1] ⊂ R. To this end both analytic and numerical methods are employed. New high-accuracy spectral data are obtained. A direct analytic proof is given that trigonometric functions cos(nπx/2) and sin(nπx), for integer n are not the eigenfunctions of |∆| 1/2 D , D = (−1, 1). This clearly demonstrates that the traditional Fourier multiplier representation of |∆| 1/2 becomes defective, while passing from R to a bounded spatial domain D ⊂ R.

Some materials can have the dispersionless parts in their electronic spectra. These parts are usually called flat bands and generate the corps of unusual physical properties of such materials. These flat bands are induced by the condensation of fermionic quasiparticles [1-3], being very similar to the Bose condensation. The difference is that fermions to condense, the Fermi surface should change its topology [1-4], leading to violation of time-reversal (T) and particle-hole (C) symmetries. Thus, the famous Landau theory of Fermi liquids does not work for the systems with fermion condensate (FC) so that several experimentally observable anomalies have not been explained so far.Here we use FC approach to explain recent observations of the asymmetric tunneling conductivity in heavy-fermion compounds and graphene [5][6][7] and its restoration in magnetic fields, as well as the violation of Leggett theorem [8,9], recently observed experimentally [10,11] in overdoped cuprates, and recent observation of the challenging universal scaling connecting linear-T -dependent resistivity to the superconducting superfluid density [12].

We study the joint effect of disorder and Coulomb interaction screening on the exciton spectra in two-dimensional (2D) structures. These can be van der Waals structures or heterostructures of organic (polymeric) semiconductors as well as inorganic substances like transition metal dichalcogenides. We consider 2D screened hydrogenic problem with Rytova–Keldysh interaction by means of so-called fractional Scrödinger equation. Our main finding is that above synergy between screening and disorder either destroys the exciton (strong screening) or promote the creation of a bound state, leading to its collapse in the extreme case. Our second finding is energy levels crossing, i.e. the degeneracy (with respect to index $$\mu $$ μ ) of the exciton eigenenergies at certain discrete value of screening radius. Latter effects may also be related to the quantum manifestations of chaotic exciton behavior in above 2D semiconductor structures. Hence, they should be considered in device applications, where the interplay between dielectric screening and disorder is important.