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General principles for describing and parametrizing complex nanoobjects or nanosystems are examined which will retain the essential information in a small number of parameters. In different situations, this is done by interpolation over spatial parameters, correlation of properties over a set of nanoobjects of the same type, abbreviated descriptions invoking effective quantum numbers, and modern approximation techniques for obtaining interpolation formulas.

The ensemble of possible and existing nano-objects and nanomaterials is treated as a multilevel multidimensional system. We introduce the quantitative and qualitative characteristics of elements of the system and give a graphical representation for it. A systems approach makes it possible to establish a basis for a continuously updateable and refineable database including the important properties of nano-objects and their interconnections. Creation of such a database is expedient for dynamic data support of nanometrology in its fundamental and applied aspects.

Features of the metrology of quantum multiparticle systems in connection with general system theory and representational theory are considered. Quantitative characteristics of the periodic system of elements (PSE) are introduced and studied, including property correlation, and also system hierarchies are considered leading to its analysis and possible application of nanosystems and quantum computers.

We consider a class of potentials for which the exact semiclassical quantization is achieved by a certain modification of the quantization condition. A list of potentials for which the new quantization condition is exact coincides with the list of potentials for which the spectrum is determined by the factorization method. We construct a one-parameter family of quantization conditions including the supersymmetric WKB condition as a special case. The new condition allows considering the interrelations between different modifications of the leading approximation and their validity ranges and also allows developing new approximate methods for calculating spectra.

With regard to applications in quantum theory, we consider the classical wave equation involving the scalar curvature with an arbitrary coefficient ξ. General properties of this equation and its solutions are studied based on modern results in group analysis with the aim to fix a physically justified value of ξ. These properties depend essentially not only on the values of ξ and the mass parameter but also on the type and dimension of the space. Form invariance and conformal invariance must be distinguished in general. A class of Lorentz spaces in which the massless equation satisfies the Huygens principle and its Green's function is free of a logarithmic singularity exists only for the conformal value of ξ. The same value of ξ follows from other arguments and the relation to the known WKB transformation method that we establish.

The current view is presented on the quantitative description of the periodic system, which is a multiparticle quantum system. Principles are proposed for extending the classification of such systems.

We develop a new approach to semiclassical quantization conditions that does not involve asymptotic expansions and is based on exact general properties of wave equations and their spectra. For centrally symmetric potentials, the quantization conditions depend only on a collection of integrals involving powers of the classical momentum. The energy levels calculated using these conditions are in good agreement with numerical data.

We develop a new version of the semiclassical analysis of a system of bound states in centrally symmetrical potentials. The set of potentials is in a 1∶1 correspondence with a certain set of pairs of functions of the orbital momentum. The first of these functions determines the usual WKB quantization condition and groups the potentials into equivalence classes. Its Mellin transform demonstrates similar behavior for the typical potentials, which allows describing the equivalence class with a small number of parameters. We can chose these parameters as the asymptotically exact estimates of the number of states. We obtain an equation that allows classifying states in a self-consistent atomic potential without knowing the explicit form of the potential. The second of these functions distinguishes the potentials within an equivalence class and also gives the first correction to the quantization condition.

The inverse problem of recovering an effective atomic potential that is universal for all elements of the periodic system is examined using a refined WKB method. The order of filling the shells is shown to be given by a linear combination of the principal and orbital quantum numbers with a coefficient that depends only on the asymptotic behavior of the potential. A relationship between the atomic numbers of the elements which open the shells and a certain functional (the strength) of the potential is found. Narrow ranges for the asymptotic index and strength of the potential are determined by comparison with data from the actual periodic system. The self consistency of this approach is demonstrated by good agreement between the calculated and actual atomic numbers of the elements.

A study has been made of the effect of superheating of the melt on the position of the “solidification plateau” of tin. It is shown that crystallization occurs in a stepwise manner. An explanation is proposed in terms of coordinated atomic spheres, by the presence of short- and long-range bonds and their anisotropy.

An effective quantum number as some linear combination of the radial and orbital quantum numbers with the coefficient depending on the specific form of centrally symmetric potentials is introduced. This quantum number defines the order of the level succession and the values of their energies by means of a modified semiclassical quantization condition. The developing method presents some perturbation theory treating given potentials in comparison with the oscillator and Coulomb ones.

An effective quantum number that determines with high accuracy the level ordering in arbitrary centrally symmetric potentials for any space dimensionality is introduced and calculated by means of certain universal methods based on the known estimates for the total number of bound states in the same potential for various dimensionalities. Our new approach reproduces many known exact results. The effective number is used for constructing the periodic system of the atomic electron shells.

A new method is proposed for calculating the vacuum energy-momentum tensor on bounded manifolds that does not require a cutoff or other artificial devices. The impenetrable boundaries are replaced by a family of localized potentials which depend on a parameter λ, the boundary condition corresponding to impenetrability being obtained in the limit λ → ∞. The Casimir effect in the case of two parallel plates is considered in detail.

The analytic dependences of the vacuum averages of the energy-momentum tensor on geometry and topology are found for scalar, electromagnetic, and massless gluon fields in bounded three-dimensional manifolds. Application of the results obtained to the “bag” model in the theory of hadrons is discussed.

A method is proposed for computing corrections to the Casimir force caused by a broken symmetry of the boundaries. The values of these corrections are obtained for a parallelepiped in the case of scalar and electromagnetic fields.

We consider the topological and geometric relations of the energy-momentum tensor of zero-point vacuum vibrations of a scalar field in two-dimensional manifolds. It is shown that the energy density can change sign under continuous deformation of the region.

Recent experiments have shown that there are discrepancies in Avagadro's number of the order of 10/sup /minus/6/, which are much greater than the errors in deriving the lattice constants in individual experiments (about 10/sup /minus/8/) on Si. This makes it important to examine possible residual systematic errors, in particular the error due to structure imperfection in the Si crystal and surface imperfections in the standard bodies, in addition to specific error sources arising from the methods of experiment. These features together define the limiting-constraint level. This paper considers the limiting-constraint features of residual stresses and strains and crystal response to external pressure as well as solid-solution features for a silicon crystal grown by crucible-free zone melting.

On an example of a two-band system with the electron-phonon interaction we show that a homogeneous stationary spontaneous current may arise when a phase transition in the presence of an external electromagnetic field occurs. The current is proportional to the imaginary part of the vector order parameter. In particular, in an illuminated homogeneous perfect ferroelectric this current appears simultaneously with the spontaneous polarization. The contribution of the spontaneous current into the photogalvanic (photovoltaic) effect is traced.

An expression for the temperature of a spherically symmetric uncharged black hole is found with the help of the general technique proposed in [8] for determining the effective vacuum temperature in external fields, based on investigation of the local properties of the Green function.