Monotonicity formulae and Liouville theorems of harmonic maps with potential

We use the stress–energy tensor to obtain the monotonicity formula and vanishing theorems for harmonic maps with potential H$H$ under some conditions on H$H$. We also obtain the unique constant solution of the constant Dirichlet boundary value problem on some starlike domain for harmonic maps with potential.     

Variational calculations on the energy levels of graphene quantum antidots

The critical properties of the two-dimensional Ising and Blume-Capel model on directedsmall-world lattices with quenched connectivity disorder are investigated. The disordered system is simulated by applying the Monte Carlo method with heat bath update algorithm and histogram re-weighting techniques. The critical temperature, as well as the critical exponents are obtained. For both models the critical parameters have been obtained for several values of the rewiring probability p. It is found that these disorder systems do not belong to the same universality class as two-dimensional ferromagnetic model on regular lattices. In particular, the Blume-Capel model, with zero crystal field interaction, on a directedsmall-world lattice presents a second-order phase transition for p < p _c , and a first-order phase transition for p > p _c , where p _c  ≈ 0.25. The critical exponents for p < p _c are different from those of the same model on a regular lattice, but are identical to the exponents of the Ising model on directedsmall-world lattice.     

Relative grain boundary free energy and surface free energy of some metals and alloys

By the method of thermal etching measurements were carried out of the ratio of grain boundary free energy _GB to the surface free energy _s in silver, copper, nickel, gamma-iron and cobalt of 99·999 pct. purity each, three copper-aluminium alloys and eight nickel-cobalt alloys, the total concentration of impurities in each alloy not exceeding 0·01 pct, as a function of the temperature and sometimes of the annealing medium.     

Higher surface energy of free nanoparticles

We present an accurate online method for the study of size-dependent evaporation of free nanoparticles allowing us to detect a size change of 0.1 nm. This method is applied to Ag nanoparticles. The linear relation between the onset temperature of evaporation and the inverse of the particle size verifies the Kelvin effect and predicts a surface energy of 7.2 J/m(2) for free Ag nanoparticles. The surface energy of nanoparticles is significantly higher as compared to that of the bulk and is essential for processes such as melting, coalescence, evaporation, growth, etc., of nanoparticles.     

The mass-shell and gauge-fixing conditions for the free relativistic massive quantum particle

Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR ⁰,R ⁺, andD as matrix element conditions to be satisfied by the on-mass-shell states of a massive particle. We evaluate the matrix elements inp-space using five sets of co-ordinates: (p ²,p), (p ²,p ⁺,p _T ), (p ²,p ^?,p _T ), (p ²,π), and (p ²,π ⁺,π _T ) where \(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \) . We find that, only in the case ofR ⁰ with (p ²,p) coordinates,R ⁺ with (p ²,p ⁺,p _T ) coordinates, andD with (p ², π) or (p ²,π ⁺,π _T ) coordinates, can the condition be satisfied by arbitrary on-mass-shell states. In all other cases, the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass-shell Hilbert space, i.e it forces a violation of the superposition principle. These results constitute thep-space quantum version of Shanmugadhasan's theorem for constrained classical systems which states that there exists, at least locally in phase space, a canonical transformation to a set of variables in which the second-class constraints become canonical pairs equal to zero with the other canonical coordinates independent of the second-class constraints.     

Thermodynamic anomaly of the free damped quantum particle: the bath perspective

By varying the orientation of the applied magnetic field with respect to the normal of a two-dimensional electron gas, the chemical potential and the specific heat reveal persistent spin splitting in all field ranges. The corresponding shape of the thermodynamic quantities distinguishes whether the Rashba spin-orbit interaction (RSOI), the Zeeman term or both dominate the splitting. The interplay of the tilting of the magnetic field and RSOI resulted to an amplified splitting even in weak fields. The effects of changing the RSOI strength and the Landau level broadening are also investigated.     

The free energy of quantum spin systems and large deviations

For a quantum system ofn identical spins of magnitudej, we introduce an integrated density of states of definite total spin angular momentum. The underlying sequence of probability measures satisfies Varadhan's large deviation principle, and converges to a degenerate distribution. We use the Berezin-Lieb Inequalities to obtain upper and lower bounds for the limiting specific free-energy of the spins interacting with a second quantum system under specified conditions on the Hamiltonian. The method is illustrated by applications to the BCS model and to the Dicke maser model.     

First order quantum correction to the free energy of simple liquids

Using the perturbation theory of Weekset al the first order quantum correction to the free energy of a simple fluid characterised by a double Yukawa potential function has been expressed in a simple closed analytical form which allows numerical calculation simply on a desk calculator.     

Electronic energy levels in semiconductor quantum wells and superlattices

We present the results of the calculations of some electronic properties of semiconductor quantum wells and superlattices. The review includes the superlattice band structure and the quantum well bound energy levels; the virtual bound states of semiconductor quantum wells and their influence on the energy spectrum of separate confinement heterostructures. Finally the perturbation of quantum well bound states and exciton states by a static electric field applied parallel to the growth axis is considered.     

Nonperturbative weak-coupling analysis of the quantum Liouville field theory

Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Bäcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g¹⁰ using both approaches, yielding identical results.     

Effects of crystal shape on the energy levels of zero-dimensional PbS quantum dots

Nanometer-size PbS quantum dots have been made by electrodeposition on a Au(111) substrate. The deposited nanocrystals have a flattened cubic shape. We probed the single-electron energy-level spectrum of individual quantum dots by scanning tunneling spectroscopy and found that it deviates strongly from that of spherical PbS quantum dots. The measured energy-level spectrum is successfully explained by considering strong confinement in a flattened cubic box.     

On the surface contribution to the grand-canonical pressure of free quantum gases

The surface term in the thermodynamic pressure of free quantum gases is proved to exist and is evaluated. Detailed proofs are given for parallelepipedic domains with Dirichlet, periodic, and Neumann boundary conditions and for more general domains with Dirichlet boundary conditions.     

Analyticity properties of the surface free energy of the Ising model

For an Ising ferromagnet with nearest-neighbour interactions of strengthK and surface magnetic fieldh, the surface free energy in the presence of a positively (or negatively) magnetized zero-field bulk phase is shown to be analytic inh for Reh<K–/, where =2.96 ... and is the inverse temperature. This puts the lower boundK–/ on the values ofh at which wetting and layering transitions can take place.     

Novel local free energy minimum on the Cu(001) surface

High-resolution LEED (low-energy electron diffraction) data of Cu(001) reveal an uniaxial in-plane lattice reconstruction by 1%. One-dimensional nanogrooves induced by ion bombardment involve the creation of steps that enable this reconstruction. This is the first verification of van der Merwe's prediction of step facilitated reconstruction. We confirm the predicted dependence on step orientation: <100> steps allow stress-relief and <110> steps do not, consistent with the known elastic anisotropy. Similar behavior is predicted for other nonreconstructed (001) surfaces of 3d and 4d metals.     

An exact operator solution of the quantum Liouville field theory

Exact, closed form results are given expressing the quantum Liouville field theory in terms of a canonical free pseudoscalar field. The classical conformal transformation properties and a Bäcklund transformation of the Liouville model are briefly reviewed and then developed into explicit operator statements for the quantum theory. This development leads to exact expressions for the basic operator functions of the Liouville field: ?_μΦ, and e^nΦ. An operator product analysis is then used to construct the Liouville energy-momentum tensor operator, which is shown to be equal to that of a free pseudoscalar field. Dynamical consequences of this equivalence are discussed, including the relation between the Liouville and free field energy eigenstates. Liouville correlation functions are partially analyzed, and remaining open questions are discussed.     

Representation of the excited energy levels of quantum systems via the functional integrals

It is shown that an arbitrary excited energy level of a quantum system can be represented in the compact form via the functional integrals. The well-known formulae for the ground state energy are generalized, thus enlarging the application area of the functional integrals in quantum mechanics.