Questo testo, giunto alla seconda edizione, è adatto per una prima esposizione della teoria delle funzioni di singola variabile complessa e si rivolge a studenti di Fisica, Matematica e Ingegneria che abbiano acquisito le nozioni fondamentali dell'Analisi Matematica reale. L esigenza di una nuova pubblicazione nasce dall idea di effettuare una selezione di argomenti, ritenuti fondamentali, la cui esposizione risulti sistematica e autoconsistente in circa 60 ore di lezione mantenendo, al tempo stesso, il rigore matematico volto a favorire la maturazione scientifica dello studente e prepararlo alla lettura di testi avanzati. A corredo della trattazione teorica vengono proposti circa 250 esercizi, raccolti tra le prove scritte assegnate per il superamento del corso, tutti forniti di soluzione dettagliata. Il loro svolgimento costituisce una parte imprescindibile per l'acquisizione della materia.

Questo testo è adatto per una prima esposizione della teoria delle funzioni di singola variabile complessa. Esso si rivolge a studenti di Fisica, Matematica e Ingegneria che abbiano acquisito le nozioni fondamentali dell'Analisi Matematica reale. L'esigenza di una nuova pubblicazione nasce dall'idea di effettuare una selezione di argomenti, ritenuti fondamentali, la cui esposizione risulti sistematica e autoconsistente in circa 60 ore di lezione mantenendo, al tempo stesso, il rigore matematico volto a favorire la maturazione scientifica dello studente e prepararlo alla lettura di testi avanzati. A corredo della trattazione teorica, vengono proposti oltre 200 esercizi tutti forniti di soluzione dettagliata. Il loro svolgimento costituisce una parte imprescindibile per l'acquisizione della materia.

We analyze a class of quantum Feller semigroups describing the evolution of interacting quantum lattice systems, specified either as generic qudits or as fermions. The corresponding generators, which include both conservative and dissipative evolutions, are given by the superposition of local generators in the Lindblad form. The associated infinite volume dynamics can be obtained as the strong limit of the finite volume dynamics. By regarding the interacting evolution as a perturbation of a non-interacting dissipative dynamics, we obtain a quantitative criterion that yields the uniqueness of the stationary state together with the exponential convergence of local observables. The analysis is based on suitable a priori bounds on the resolvent equation which yield quantitive estimates on the evolution of local observables.

We discuss conditions for the enhancement of fusion reactivities arising from different choices of energy distribution functions for the reactants. The key element for potential gains in fusion reactivity is identified in the functional dependence of the tunneling coefficient on the energy, ensuring the existence of a finite range of temperatures for which reactivity of fusion processes is boosted with respect to the Maxwellian case. This is shown using a convenient parametrization of the tunneling coefficient dependence on the energy, analytically in the simplified case of a bimodal Maxwell-Boltzmann distribution, and numerically for kappa distributions. We then consider tunneling potentials progressively better approximating fusion processes and evaluate in each case the average reactivity in the case of kappa distributions.

We discuss the sensitivity of tunneling processes to the initial preparation of the quantum state. We compare the case of Gaussian wave packets of different positional variances using a generalized Woods-Saxon potential for which analytical expressions of the tunneling coefficients are available. Using realistic parameters for barrier potentials we find that the usual plane wave approximation underestimates fusion reactivities by an order of magnitude in a range of temperatures of practical relevance for controlled energy production.

In nature, everything occurs at finite temperature, and quantum phase transitions (QPTs) cannot be an exception. Nevertheless, they are still mainly discussed and formulated at zero temperature. We show that the condensation QPTs recently introduced at zero temperature can naturally be extended to finite temperature just by replacing ground state energies with corresponding free energies. We illustrate this criterion in the paradigmatic Grover model and in a system of free fermions in a one-dimensional inhomogeneous lattice. In agreement with expected universal features, the two systems show structurally similar phase diagrams. Last, we explain how finite temperature condensation QPTs can be used to construct quantum annealers having, at finite temperature, output-probability exponentially close to 1 in the system size. As examples we consider again the Grover model and the fermionic system, the latter being well within the reach of present heterostructure technology.

We formalize and prove the extension to finite temperature of a class of quantum phase transitions, acting as condensations in the space of states, recently introduced and discussed at zero temperature (Ostilli and Presilla 2021 J. Phys. A: Math. Theor. 54 055005). In details, we find that if, for a quantum system at canonical thermal equilibrium, one can find a partition of its Hilbert space ℋ into two subspaces, ℋ_cond and ℋ_norm, such that, in the thermodynamic limit, dim ℋ_cond / dim ℋ_norm → 0 and the free energies of the system restricted to these subspaces cross each other for some value of the Hamiltonian parameters, then, the system undergoes a first-order quantum phase transition driven by those parameters. The proof is based on an exact probabilistic representation of quantum dynamics at an imaginary time identified with the inverse temperature of the system. We also show that the critical surface has universal features at high and low temperatures.

When noninteracting fermions are confined in a D-dimensional region of volume O(L^D) and subjected to a continuous (or piecewise-continuous) potential V which decays sufficiently fast with distance, in the thermodynamic limit, the ground-state energy of the system does not depend on V. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in D dimensions. We find that this universality property holds under a quite mild condition on V, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.

We study the ground state of a system of spinless electrons interacting through a screened Coulomb potential in a lattice ring. By using analytical arguments, we show that, when the effective interaction compares with the kinetic energy, the system forms a Wigner crystal undergoing a first-order quantum phase transition. This transition is a condensation in the space of the states and belongs to the class of quantum phase transitions discussed in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021)]. The transition takes place at a critical value r_sc of the usual dimensionless parameter r_s (radius of the volume available to each electron divided by effective Bohr radius) for which we are able to provide rigorous lower and upper bounds. For large screening length these bounds can be expressed in a closed analytical form. Demanding Monte Carlo simulations allow to estimate r_sc = 2.3 ± 0.2 at lattice filling 3/10 and screening length 10 lattice constants. This value is well within the rigorous bounds 0.7 < r_sc < 4.3. Finally, we show that if screening is removed after the thermodynamic limit has been taken, r_sc tends to zero. In contrast, in a bare unscreened Coulomb potential, Wigner crystallization always takes place as a smooth crossover, not as a quantum phase transition.

We consider an open quantum system with dissipation, described by a Lindblad Master equation (LME). For dissipation locally acting and sufficiently strong, a separation of the relaxation timescales occurs, which, in terms of the eigenvalues of the Liouvillian, implies a grouping of the latter in distinct vertical stripes in the complex plane at positions determined by the eigenvalues of the dissipator. We derive effective LME equations describing the modes within each stripe separately, and solve them perturbatively, obtaining for the full set of eigenvalues and eigenstates of the Liouvillian explicit expressions correct at order 1/Γ included, where Γ is the strength of the dissipation. As an example, we apply our general results to quantum XYZ spin chains coupled, at one boundary, to a dissipative bath of polarization.

We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian H=K+gV, where K and V are two non commuting operators acting on the space of states 𝔽, we may always write 𝔽=𝔽_cond ⊕ 𝔽_norm where 𝔽_cond is the subspace spanned by the eigenstates of V with minimal eigenvalue and 𝔽_cond= 𝔽_cond^⊥. If, in the thermodynamic limit, M_cond / M → 0, where M and M_cond are, respectively, the dimensions of 𝔽 and 𝔽_cond, the above decomposition of 𝔽 becomes effective, in the sense that the ground state energy per particle of the system, ε, coincides with the smaller between ε_cond and ε_norm, the ground state energies per particle of the system restricted to the subspaces 𝔽_cond and 𝔽_norm, respectively: ε=min{ε_cond, ε_norm}. It may then happen that, as a function of the parameter g, the energies ε_cond and ε_norm cross at g=g_c. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace 𝔽_cond) and a normal phase (system spread over the large subspace 𝔽_norm). Since, in the thermodynamic limit, M_cond / M → 0, the confinement into 𝔽_cond is actually a condensation in which the system falls into a ground state orthogonal to that of the normal phase, something reminiscent of Anderson's orthogonality catastrophe (P.W. Anderson, Phys. Rev. Lett. 18, 1049 (1967)). The outlined mechanism is tested on a variety of benchmark lattice models, including spin systems, free fermions with non uniform fields, interacting fermions and interacting hard-core bosons.

We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.

By using a previously established exact characterization of the ground state of random potential systems in the thermodynamic limit, we determine the ground and first excited energy levels of quantum random energy models, discrete and continuous. We rigorously establish the existence of a universal first order quantum phase transition, obeyed by both the ground and the first excited states. The presence of an exponentially vanishing minimal gap at the transition is general but, quite interestingly, the gap averaged over the realizations of the random potential is finite. This fact leaves still open the chance for some effective quantum annealing algorithm, not necessarily based on a quantum adiabatic scheme.

We investigate the time evolution of an open quantum system described by a Lindblad master equation with dissipation acting only on a part of the degrees of freedom H0 of the system, and targeting a unique dark state in H₀. We show that, in the Zeno limit of large dissipation, the density matrix of the system traced over the dissipative subspace H₀, evolves according to another Lindblad dynamics, with renormalized effective Hamiltonian and weak effective dissipation. This behavior is explicitly checked in the case of Heisenberg spin chains with one or both boundary spins strongly coupled to a magnetic reservoir. Moreover, the populations of the eigenstates of the renormalized effective Hamiltonian evolve in time according to a classical Markov dynamics. As a direct application of this result, we propose a computationally-efficient exact method to evaluate the nonequilibrium steady state of a general system in the limit of strong dissipation.

We investigate the non-equilibrium steady state (NESS) in an open quantum XXZ chain attached at the ends to polarization baths with unequal polarizations. Using the general theory developed in Popkov (2017 Phys. Rev. A 95, 052131), we show that in the critical XXZ |Δ|<1 easy plane case, the steady current in large systems under strong driving shows resonance-like behaviour, by an infinitesimal change of the spin chain anisotropy or other parameters. Alternatively, by fine tuning the system parameters and varying the boundary dissipation strength, we observe a change of the NESS current from diffusive (of order 1/N, for small dissipation strength) to ballistic regime (of order 1, for large dissipation strength). This drastic change results from an accompanying structural change of the NESS, which becomes a pure spin-helix state characterized by a winding number which is proportional to the system size. We calculate the critical dissipation strength needed to observe this surprising effect.

We study the thermalization of an ensemble of N elementary, arbitrarily-complex, quantum sys- tems, mutually noninteracting but coupled as electric or magnetic dipoles to a blackbody radiation. The elementary systems can be all the same or belonging to different species, distinguishable or indistinguishable, located at fixed positions or having translational degrees of freedom. Even if the energy spectra of the constituent systems are nondegenerate, as we suppose, the ensemble unavoid- ably presents degeneracies of the energy levels and/or of the energy gaps. We show that, due to these degeneracies, a thermalization analysis performed by the popular quantum optical master equation reveals a number of serious pathologies, possibly including a lack of ergodicity. On the other hand, a consistent thermalization scenario is obtained by introducing a Lindblad-based approach, in which the Lindblad operators, instead of being derived from a microscopic calculation, are established as the elements of an operatorial basis with squared amplitudes fixed by imposing a detailed balance condition and requiring their correspondence with the dipole transition rates evaluated at the first order perturbation theory. Due to the above mentioned degeneracies, this procedure suffers a basis arbitrariness which, however, can be removed exploiting the fact that the themalization of an en- semble of equal noninteracting systems cannot rely on the ensemble size. As a result, we provide a clear-cut partitioning of the thermalization time into dissipation and decoherence times, for which we derive formulas giving the dependence on the energy levels of the elementary systems, the size N of the ensemble and the temperature of the blackbody radiation.

We propose a solution to the problem of realizing a predefined and arbitrary pure quantum state, based on the simultaneous presence of coherent and dissipative dynamics, noncommuting on the target state and in the limit of strong dissipation. More precisely, we obtain a necessary and sufficient criterion whereby the nonequilibrium steady state (NESS) of an open quantum system described by a Lindblad master equation approaches a target pure state in the Zeno regime, i.e., for infinitely large dissipative coupling. We also provide an explicit formula for the characteristic dissipative strength beyond which the purity of the NESS becomes effective, thus paving the way to an experimental implementation of our criterion. For an illustration, we deal with targeting a Bell state, an arbitrary pure state of N qubits, and a spin-helix state of N qubits.

In a recent work, we have derived simple Lindblad-based equations for the thermalization of systems in contact with a thermal reservoir. Here, we apply these equations to the Lipkin-Meshkov-Glick model in contact with blackbody radiation and analyze the dipole matrix elements involved in the thermalization process. We find that the thermalization can be complete only if the density is sufficiently high, while, in the limit of low density, the system thermalizes partially, namely, within the Hilbert subspaces where the total spin has a fixed value. In this regime, and in the isotropic case, we evaluate the characteristic thermalization time analytically, and show that it diverges with the system size in correspondence with the critical points and inside the ferromagnetic region. Quite interestingly, at zero temperature the thermalization time diverges only quadratically with the system size, whereas quantum adiabatic algorithms, aimed at finding the ground state of the same system, imply a cubic divergence of the required adiabatic time.

We review the calculation of Fermi's golden rule for a system of N-body dipoles, magnetic or electric, weakly interacting with a blackbody radiation. By using the magnetic or electric field-field correlation function evaluated in the 1960s for the black-body radiation, we deduce a general formula for the transition rates and study its limiting, fully coherent or fully incoherent, regimes.

Dissipative preparation of a pure steady state usually involves a commutative action of a coherent and a dissipative dynamics on the target state. Namely, the target pure state is an eigenstate of both the coherent and dissipative parts of the dynamics. We show that working in the Zeno regime, i.e., for infinitely large dissipative coupling, one can generate a pure state by a noncommutative action, in the above sense, of the coherent and dissipative dynamics. A corresponding Zeno regime pureness criterion is derived. We illustrate the approach, looking at both its theoretical and applicative aspects, in the example case of an open XXZ spin-1/2 chain, driven out of equilibrium by boundary reservoirs targeting different spin orientations. Using our criterion, we find two families of pure nonequilibrium steady states, in the Zeno regime, and calculate the dissipative strengths effectively needed to generate steady states which are almost indistinguishable from the target pure states.

A possible solution of the well known paradox of chiral molecules is based on the idea of spontaneous symmetry breaking. At low pressure the molecules are delocalized between the two minima of a given molecular potential while at higher pressure they become localized in one minimum due to the intermolecular dipole-dipole interactions. Evidence for such a phase transition is provided by measurements of the inversion spectrum of ammonia and deuterated ammonia at different pressures. In particular, at pressure greater than a critical value no inversion line is observed. These data are well accounted for by a model previously developed and recently extended to mixtures. In the present paper, we discuss the variation of the critical pressure in binary mixtures as a function of the fractions of the constituents.

According to quantum mechanics chiral molecules, that is molecules that rotate the polarization of light, should not exist. The simplest molecules which can be chiral have four or more atoms with two arrangements of minimal potential energy that are equivalent up to a parity operation. Chiral molecules correspond to states localized in one potential energy minimum and can not be stationary states of the Schrödinger equation. A possible solution of the paradox can be founded on the idea of spontaneous symmetry breaking. This idea was behind work we did previously involving a localization phase transition: at low pressure the molecules are delocalized between the two minima of the potential energy while at higher pressure they become localized in one minimum due to the intermolecular dipole-dipole interactions. Evidence for such a transition is provided by measurements of the inversion spectrum of ammonia and deuterated ammonia at different pressures. A previously proposed model gives a satisfactory account of the empirical results without free parameters. In this paper, we extend this model to gas mixtures. We find that also in these systems a phase transition takes place at a critical pressure which depends on the composition of the mixture. Moreover, we derive formulas giving the dependence of the inversion frequencies on the pressure. These predictions are susceptible to experimental test.

In the probabilistic approach to quantum many-body systems, the ground-state energy is the solution of a nonlinear scalar equation written either as a cumulant expansion or as an expectation with respect to a probability distribution of the potential and hopping (amplitude and phase) values recorded during an infinitely lengthy evolution. We introduce a perturbative expansion of this probability distribution which conserves, at any order, a multinomial-like structure, typical of uncorrelated systems, but includes, order by order, the statistical correlations provided by the cumulant expansion. The proposed perturbative scheme is successfully tested in the case of pseudo-spin 1/2 hard-core boson Hubbard models also when affected by a phase problem due to an applied magnetic field.

We discuss the cooling efficiency of ultracold Fermi-Bose mixtures in species-selective traps using a thermodynamical approach. The dynamics of evaporative cooling trajectories is analyzed in the specific case of bichromatic optical dipole traps also taking into account the effect of partial spatial overlap between the Fermi gas and the thermal component of the Bose gas. We show that large trapping frequency ratios between the Fermi and the Bose species allow for the achievement of a deeper Fermi degeneracy, consolidating in a thermodynamic setting earlier arguments based on more restrictive assumptions. In particular, we confirm that the minimum temperature of the mixture is obtained at the crossover between boson and fermion heat capacities, and that below such a temperature sympathetic cooling vanishes. When the effect of partial overlap is taken into account, optimal sympathetic cooling of the Fermi species may be achieved by properly tuning the relative trapping strength of the two species in a time-dependent fashion. Alternatively, the dimensionality of the trap in the final stage of cooling can be changed by increasing the confinement strength, which also results in a crossover of the heat capacities at deeper Fermi degeneracies. This technique may be extended to Fermi-Bose degenerate mixtures in optical lattices.

By using a recently proposed probabilistic approach, we determine the exact ground state of a class of matrix Hamiltonian models characterized by the fact that in the thermodynamic limit the multiplicities of the potential values assumed by the system during its evolution are distributed according to a multinomial probability density. The class includes (i) the uniformly fully connected models, namely a collection of states all connected with equal hopping coefficients and in the presence of a potential operator with arbitrary levels and degeneracies, and (ii) the random potential systems, in which the hopping operator is generic and arbitrary potential levels are assigned randomly to the states with arbitrary probabilities. For this class of models we find a universal thermodynamic limit characterized only by the levels of the potential, rescaled by the ground-state energy of the system for zero potential, and by the corresponding degeneracies (probabilities). If the degeneracy (probability) of the lowest potential level tends to zero, the ground state of the system undergoes a quantum phase transition between a normal phase and a frozen phase with zero hopping energy. In the frozen phase the ground state condenses into the subspace spanned by the states of the system associated with the lowest potential level.

The recent remarkable developments in quantum optics, mesoscopic and cold atom physics have given reality to wave functions. It is then interesting to explore the consequences of assuming ensembles over the wave functions simply related to the canonical density matrix. In this note we analyze a previously introduced distribution over wave functions which naturally arises considering the Schrödinger equation as an infinite dimensional dynamical system. In particular, we discuss the low temperature fluctuations of the quantum expectations of coordinates and momenta for a particle in a double well potential. Our results may be of interest in the study of chiral molecules.

We review a novel approach to evaluate the ground-state properties of many-body lattice systems based on an exact probabilistic representation of the dynamics and its long time approximation via a central limit theorem. The choice of the asymptotic density probability used in the calculation is discussed in detail.

We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy as well as the correlation functions in the ground state are exactly determined as series expansions in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of the analytic character of the present approach is discussed.

On the basis of a Feynman-Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.

We propose a double-well configuration for optical trapping of ultracold two-species Fermi-Bose atomic mixtures. Two signatures of macroscopic quantum coherence attributable to a superfluid phase transition for the Fermi gas are analyzed. The first signature is based upon tunneling of Fermi pairs when the power of the deconfining laser beam is significantly reduced. The second relies on the observation of interference fringes in a regime where the fermions are trapped in two sharply separated minima of the potential. Both signatures rely on small decoherence times for the Fermi samples, which should be possible by reaching low temperatures using a Bose gas as a refrigerator, and a bichromatic optical dipole trap for confinement, with optimal heat-capacity matching between the two species.

On the grounds of a Feynman-Kac-type formula for Hamiltonian lattice systems, we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result, we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case.

In a previous paper we proposed a model to describe a gas of pyramidal molecules interacting via dipole-dipole interactions. The interaction modifies the tunneling properties between the classical equilibrium configurations of the single molecule and, for sufficiently high pressure, the molecules become localized in these classical configurations. The model explains quantitatively the shift to zero-frequency of the inversion line observed upon increase of the pressure in a gas of ammonia or deuterated ammonia. Here we analyze further the model especially with respect to stability questions.

The study of low density, ultracold atomic Fermi gases is a promising avenue to understand fermion superfluidity from first principles. One technique currently used to bring Fermi gases in the degenerate regime is sympathetic cooling through a reservoir made of an ultracold Bose gas. We discuss a proposal for trapping and cooling of two-species Fermi-Bose mixtures into optical dipole traps made from combinations of laser beams having two different wavelengths. In these bichromatic traps it is possible, by a proper choice of the relative laser powers, to selectively trap the two species in such a way that fermions experience a stronger confinement than bosons. As a consequence, a deep Fermi degeneracy can be reached having at the same time a softer degenerate regime for the Bose gas. This leads to an increase in the sympathetic cooling efficiency and allows for higher precision thermometry of the Fermi-Bose mixture.

We compare strategies for evaporative and sympathetic cooling of two-species Fermi-Bose mixtures in single-color and two-color optical dipole traps. We show that in the latter case a large heat capacity of the bosonic species can be maintained during the entire cooling process. This could allow one to efficiently achieve a deep Fermi degeneracy regime having at the same time a significant thermal fraction for the Bose gas, crucial for a precise thermometry of the mixture. Two possible signatures of a superfluid phase transition for the Fermi species are discussed.

We propose the use of a combined optical dipole trap to achieve Fermi degeneracy by sympathetic cooling with a different bosonic species. Two far-detuned pairs of laser beams focused on the atomic clouds are used to confine the two atomic species with different trapping strengths. We show that a deep Fermi degeneracy regime can be potentially achieved earlier than Bose-Einstein condensation, as discussed in the favorable situation of a 6Li-23Na mixture. This opens up the possibility of experimentally investigating a mixture of superfluid Fermi and normal Bose gases.

In the evolution of distributions localized around classical equilibrium points, the quantum-classical correspondence breaks down at a time, the so-called quantum breaking, or Ehrenfest time, which is related to the minimal separation of the quantum levels in proximity of the classical equilibrium energy. By studying one-dimensional systems with single- and double-well polynomial potentials, we find that the Ehrenfest time diverges logarithmically with the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. In all the other cases, we have a power law divergence with the exponent determined by the degree of the potential near the equilibrium point.

We propose a model to describe a gas of pyramidal molecules interacting via dipole-dipole interactions. The interaction modifies the tunneling properties between the classical equilibrium configurations of the single molecule and, for sufficiently high pressure, the molecules become localized in these classical configurations. We explain quantitatively, without free parameters, the shift to zero frequency of the inversion line observed upon increase of the pressure in a gas of ammonia or deuterated ammonia. For sufficiently high pressures, our model suggests the existence of a superselection rule for states of different chirality in substituted derivatives.

The stationary solutions of the Gross-Pitaevskii equation can be divided in two classes: those which reduce, in the limit of vanishing nonlinearity, to the eigenfunctions of the associated Schrödinger equation and those which do not have linear counterpart. Analytical and numerical results support an existence condition for the solutions of the first class in terms of the ratio between their proper frequency and the corresponding linear eigenvalue. For one-dimensional confined systems, we show that solutions without linear counterpart do exist in presence of a multiwell external potential. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.

We show the existence of stationary solutions of a one-dimensional Gross-Pitaevskii equation in the presence of a multiwell external potential that do not reduce to any of the eigenfunctions of the associated Schrödinger problem. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.

We propose a model to describe a gas of pyramidal molecules interacting via dipole-dipole interactions. A cooperative effect induced by the interaction modifies the tunneling properties between the classical equilibrium configurations of the single molecule. The model suggests that, for sufficiently high gas density, the molecules become localized in these classical configurations. On this basis it is possible to explain the shift and the disappearance of the inversion line observed upon increase of the pressure in a gas of ammonia or deuterated ammonia. The same mechanism also accounts for the presence of stable optical activity of certain pyramidal molecules. We discuss the concept of environment induced superselection rule which has been invoked in connection with this problem.

We derive an exact probabilistic representation for the evolution of a Hubbard model with site- and spin-dependent hopping coefficients and site-dependent interactions in terms of an associated stochastic dynamics of a collection of Poisson processes.

We study the stationary solutions of the Gross–Pitaevskii equation that reduce, in the limit of vanishing non-linearity, to the eigenfunctions of the associated Schrödinger equation. By providing analytical and numerical support, we conjecture an existence condition for these solutions in terms of the ratio between their proper frequency (chemical potential) and the corresponding linear eigenvalue. We also give approximate expressions for the stationary solutions which become exact in the opposite limit of strong non-linearity. For one-dimensional systems these solutions have the form of a chain of dark or bright solitons depending on the sign of the non-linearity. We demonstrate that in the case of negative non-linearity (attractive interaction) the norm of the solutions is always bounded for dimensions greater than one.

We describe an exact Feynman-Kac type formula to represent the dynamics of fermionic lattice systems. In this approach the real time or Euclidean time dynamics is expressed in terms of the stochastic evolution of a collection of Poisson processes. From this formula we derive a family of algorithms for Monte Carlo simulations, parametrized by the jump rates of the Poisson processes.

The present contribution describes the beginning of a systematic study of semiclassical evolutions using the effective action formalism. In the first part, after introducing the formalism of the effective action and its expansion in powers of h (loop-expansion) in the context of quantum mechanics, we concentrate on the structure of the first order corrections in h. These corrections are evaluated to the second order in the derivative expansion by two different methods. The first is based on a Euclidean approach, the second one on an adiabatic approximation in evaluating functional determinants. In the second part of the article we put the formalism at work, choosing as our case study a two-dimensional (2-D) anharmonic oscillator of the kind considered in molecular physics. The results of the simulations show that by increasing h the effective dynamics tends to regularize the classical motion and becomes qualitatively very similar to the quantum evolution provided the energy is sufficiently small.

We present a simple derivation of a Feynman-Kac-type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. This formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal algorithm can be chosen which coincides with the Green Function Monte Carlo method in the limit when the latter becomes exact.

We study the dynamics of classical and quantum systems linearly interacting with a classical environment represented by an infinite set of harmonic oscillators. The environment induces a dynamical localization of the quantum state into a generalized coherent state for which the h → 0 limit always exists and reproduces the classical motion. We describe the consequences of this localization on the behavior of a macroscopic system by considering the example of a Schröodinger cat.

During a continuous measurement, quantum systems can be described by a stochastic Schrödinger equation which, in the appropriate limit, reproduces the von Neumann wave-function collapse. The average behavior on the ensemble of all measurement results is described by a master equation obtained from a general model of measurement apparatus consisting of an infinite set of degrees of freedom linearly interacting with the measured system and in contact with a reservoir at high temperature.

We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and non-selective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behaviour is always recovered in the macroscopic limit.

An adiabatic approximation in terms of instantaneous resonances is developed to study the steady-state and time-dependent transport of interacting electrons in biased resonant-tunneling heterostructures. The resulting model consists of quantum reservoirs coupled to regions where the system is described by nonlinear ordinary differential equations and has a general conceptual interest.

By using numerical simulations, we investigate the dynamics of a quantum system of interacting bosons. We find an increase of properly defined mixing properties when the number of particles increases at constant density or the interaction strength drives the system away from integrability. A correspondence with the dynamical chaoticity of an associated c-number system is then used to infer properties of the quantum system in the thermodynamic limit.

An adiabatic approximation in terms of instantaneous resonances to study the steady-state and time-dependent transport properties of interacting electrons in biased resonant tunneling heterostructures is used. This approach leads, in a natural way, to a transport model of large applicability consisting of reservoirs coupled to regions where the system is described by a nonlinear Schrödinger equation. From the mathematical point of view, this work is nonrigorous but may offer some fresh and interesting problems involving semiclassical approximation, adiabatic theory, nonlinear Schrödinger equations, and dynamical systems.

We present numerical evidence that in a system of interacting bosons there exists a correspondence between the spectral properties of the exact quantum Hamiltonian and the dynamical chaos of the associated mean-field evolution. This correspondence, analogous to the usual quantum-classical correspondence, is related to the formal parallel between the second quantization of the mean field, which generates the exact dynamics of the quantum N-body system, and the first quantization of classical canonical coordinates. The limit of infinite density and the thermodynamic limit are then briefly discussed.

Measurement quantum mechanics, the theory of a quantum system which undergoes a measurement process, is introduced by a loop of mathematical equivalencies connecting previously proposed approaches. The unique phenomenological parameter of the theory is linked to the physical properties of an informational environment acting as a measurement apparatus which allows for an objective role of the observer. Comparison with a recently reported experiment suggests how to investigate novel interesting regimes for the quantum Zeno effect.

We study the photoluminescence from a near-surface quantum well in the regime of ambipolar tunneling to the surface states. Under steady-state excitation an electric field develops self-consistently due to the condition of equal tunneling currents for electrons and holes. The field induces a Stark shift of the photoluminescence signal which compares well with experimental data from near-surface GaAs/AlGaAs single quantum wells.

Continuous-wave and time-resolved photoluminescence at low temperatures is investigated in near surface quantum wells, where carrier tunneling to surface states reduces the e1-hh1 excitonic emission efficiency. Analysis of the time-resolved spectra points to a behavior for strained InGaAs/GaAs interfaces different from AlGaAs/GaAs interfaces, in that tunneling occurs not only from the free-carrier continuum, but also from the exciton ground state.

A quantum measurement model based upon restricted path-integrals allows us to study measurements of generalized position in various one-dimensional systems of phenomenological interest. After a general overview of the method we discuss the cases of a harmonic oscillator, a bistable potential and two coupled systems, briefly illustrating their applications.

We propose a numerical method for evaluating eigenvalues and eigenfunctions of Schrödinger operators with general confining potentials. The method is selective in the sense that only the eigenvalue closest to a chosen input energy is found through an absolutely stable relaxation algorithm which has a rate of convergence that is infinite. In the case of bistable potentials the method allows one to evaluate the fundamental energy splitting for a wide range of tunneling rates.

The influence of continuous measurements of energy with finite accuracy is studied in various quantum systems through a restriction of the Feynman path integrals around the measurement result. The method, which is equivalent to considering an effective Schrödinger equation with a non-Hermitian Hamiltonian, allows one to study the dynamics of the wave-function collapse. A numerical algorithm for solving the effective Schrödinger equation is developed and checked in the case of a harmonic oscillator. The situations, of physical interest, of a two-level system and of a metastable quantum well are then discussed. In the first case, the Zeno inhibition observed in quantum optics experiments is recovered and extended to nonresonant transitions, and in the second case we propose an observation of the inhibition of spontaneous decay in mesoscopic heterostructures. In all the considered examples, the effect of the continuous measurement of energy is a freezing of the evolution of the system proportional to the accuracy of the measurement itself.

We present a self-consistent approach to describe ambipolar tunnelling in asymmetrical double quantum wells under steady-state excitation and extend the results to the case of tunnelling from a near-surface quantum well to surface states. The results of the model compare very well with the behaviour observed in photoluminescence experiments in InGaAs/InP asymmetric double quantum wells and in near-surface AlGaAs/GaAs single quantum wells.

Schrödinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to separate the relevant degrees of freedom by noting that in the regions where the nonlinearities are effective all states are suppressed but the metastable ones (resonances). In this way the description of the nonlinear regions is reduced to ordinary differential equations weakly coupled to standard Schrödinger equations valid in the linear regions. The idea is illustrated through the study of a prototype equation recently proposed for resonant tunneling of electrons through a double barrier heterostructure and for which nonlinear oscillations have been numerically predicted.

A many body quantum system undergoing a multiple resonant tunneling in a double barrier heterostructure is studied within a mean field approximation. The resulting nonlinear Schrödinger equation shows genuine chaos with a positive maximum Lyapunov exponent. A simplified model of the same system suggests that chaos develops only if the interaction among the particles is nonuniform in space.

The tunneling mechanism of electrons and holes to surface states from near-surface Al0.3Ga0.7As/GaAs quantum wells has been investigated by steady-state and time-resolved photoluminescence spectroscopy, near liquid-helium temperature, of the excitonic e1-hh1 transition in the well. The ensemble of the data, taken over a wide range of optical excitation levels, for various values of the tunneling-barrier thickness, and before and after passivation of the surface by hydrogen, allows a description both of the details of the tunneling mechanism and of the character and behavior of relevant surface states. The main results are summarized as follows: (i) steady-state tunneling is ambipolar, namely, separate for electrons and holes, rather than excitonic; (ii) Spicer’s advanced unified defect model for an oxidized GaAs surface, antisite-As donors as dominating surface traps, provides an appropriate description of the state distribution at the interface between AlGaAs and its oxide; (iii) hole accumulation in surface states, resulting from the nominally different unipolar tunneling probability for the two carriers (and increasing with excitation level), generates a dipole electric field across the tunneling barrier, extending into the well; (iv) hydrogenation efficiently passivates electron trapping in surface states, but not hole tunneling and the consequent generation of a surface field by illumination; (v) the experimental findings agree with a model for ambipolar tunneling based on a self-consistent quantum-mechanical approach.

A model for the quantum Zeno effect based upon an effective Schrödinger equation originated by the path-integral approach is developed and applied to a two-level system simultaneously stimulated by a resonant perturbation. It is shown that inhibition of stimulated transitions between the two levels appears as a consequence of the influence of the meter whenever measurements of energy, either continuous or pulsed, are performed at quantum level of sensitivity. The generality of this approach allows one to qualitatively understand the inhibition of spontaneous transitions as the decay of unstable particles, originally presented as a paradox of the quantum measurement theory.

We discuss a model of repeated measurements of position in a quantum system which is monitored for a finite amount of time with a finite instrumental error. In this framework we recover the optimum monitoring of a harmonic oscillator proposed in the case of an instantaneous collapse of the wave function into an infinite-accuracy measurement result. We also establish numerically the existence of an optimal measurement strategy in the case of a nonlinear system. This optimal strategy is completely defined by the spectral properties of the nonlinear system.

We propose an electromechanical transducer based on a resonant-tunnelling configuration that, with respect to the standard tunnelling transducers, allows larger tunnelling currents while using the same bias voltage. The increased current leads to a decrease of the shot noise and an increase of the momentum noise which determine the quantum limit in the system under monitoring. Experiments with micromachined test masses at 4.2 K could show dominance of the momentum noise over the Brownian noise, allowing observation of quantum-mechanical noise at the mesoscopic scale.

We use the generalized S-matrix approach to study multiple-lead coherent conductors in the case of finite applied voltages. In this framework we discuss the transverse voltage arising in a four-lead conductor with two symmetric biased leads.

A model for dealing with energy and momentum exchanges between ballistic electrons and the vacuum barrier in a tunneling probe used as an electromechanical transducer is studied and its physical significance in devices of size comparable to the mean free path of the tunneling electrons is discussed.

An extensive Auger electron spectroscopy study of the aluminum-aluminum oxide interface is presented, with the aim of identifying the behaviour of the contact zone between bulk metal and bulk oxide. Restored oxygen KLL and aluminum LVV spectra have been derived and analysed as a function of the depth of a thick aluminum sample evaporated onto an aluminum oxide crystal. The capability of the Auger electron spectroscopy to derive rather unique information on the nature of the contact zone is used to infer that the metal and oxide form small isles or domains in the interface region, so that the transition from the metal to the oxide is not characterized by a sharp surface.

We describe a quantum many-body system undergoing multiple resonant tunneling which exhibits chaotic behavior in numerical simulations of a mean-field approximation. This phenomenon, which has no counterpart in the classical limit, is due to effective nonlinearities in the tunneling process and can be observed in principle with a heterostructure.

We show applications of electron and photon tunneling transducers for detecting small displacements in resonant bar gravitational wave detectors. A single-crystal silicon bar detector of mass 150 kg instrumented with a multimode tunneling transducer may reach h=6x10^{-19} while operating continuously at room temperature.

The fundamental sources of noise in a vacuum-tunneling probe used as an electromechanical transducer to monitor the location of a test mass are examined using a first-quantization formalism. We show that a tunneling transducer enforces the Heisenberg uncertainty principle for the position and momentum of a test mass monitored by the transducer through the presence of two sources of noise: the shot noise of the tunneling current and the momentum fluctuations transferred by the tunneling electrons to the test mass. We analyze a number of cases including symmetric and asymmetric rectangular potential barriers and a barrier in which there is a constant electric field. Practical configurations for reaching the quantum limit in measurements of the position of macroscopic bodies with such a class of transducers are studied.

Application of the path-integral approach to continuous measurements leads to effective Lagrangians or Hamiltonians in which the effect of the measurement is taken into account through an imaginary term. We apply these considerations to nonlinear oscillators with use of numerical computations to evaluate quantum limitations for monitoring position in such a class of systems.

The resonant tunneling of electrons through a double barrier is analyzed from a dynamical point of view. When a self consistent potential, representing the effect of the electrostatic feedback induced by the charge trapped in the well, is taken into account, the nonlinearity of the transition process can lead to oscillations of the transmitted fluxes. This behavior is shown to depend sensitively on the energy spread of the incident electron distribution and on the intensity of the electrostatic feedback.

The resonant tunneling of electrons through a double barrier is analyzed from a dynamical point of view. When a self consistent potential, representing the effect of the electrostatic feedback induced by the charge trapped in the well, is taken into account, the nonlinearity of the transition process can lead to oscillations of the transmitted fluxes. This behavior is shown to depend sensitively on the energy spread of the incident electron distribution and on the intensity of the electrostatic feedback.

We analyze the dynamical evolution of the resonant tunneling of an ensemble of electrons through a double barrier in the presence of the self-consistent potential created by the charge accumulation in the well. The intrinsic nonlinearity of the transmission process is shown to lead to oscillations of the stored charge and of the transmitted and reflected fluxes. The dependence on the electrostatic feedback induced by the self-consistent potential and on the energy width of the incident distribution is discussed.

The one year long history of cold fusion is critically reviewed on the basis of the more recent results, in an attempt to establish the perspectives of this field.

A Comment on the Letter by S. M. Durbin and T. Gog, Phys. Rev. Lett. 63, 1304 (1989).

The fusion rates for the reactions pd, dd and pp in a solid-state environment are determined under the assumption that the fusion process is mediated by bound states of the reactant pair. An effective binding potential which accounts for lattice, electronic and pair interactions is envisaged to model plausible equilibrium and nonequilibrium conditions. Our estimates for the upper-bound fusion rates in condensed matter lie well below the actual experimental limits.

We characterize the notion of stochastic resonance for a wide class of bistable systems driven by a periodic modulation. On developing an adiabatic picture of the underlying relaxation mechanism, we show that the intensity of the effect under study is proportional to the escape rate in the absence of perturbation. The adiabatic model of stochastic resonance accounts for the role of finite damping and finite noise correlation time as well. Our predictions compare well with the results of analogue simulation.

The relaxation properties of a one-dimensional overdamped system modulated by an external periodic force are studied analytically by means of a perturbation approach. The validity of the approximations introduced is discussed in detail. The nonstationary nature of the process is illustrated by evaluating explicitly the autocorrelation function for the relaxation in a bistable potential. The predictions thus obtained are shown to compare favorably with the results of analogue simulation for the case of a quartic double-well potential. The stochastic resonance mechanism is proven to set in only when the periodic perturbation breaks the symmetry of the bistable potential.

We review the use of supersymmetric quantum mechanics in the analysis of the classical diffusion of a brownian particle in weakly binding (soft) potentials. In particular, our computations are shown to provide an accurate analytical determination of the shape-mode frequency of the static double sine-Gordon kink.

The supersymmetric connection between Fokker-Planck and Schrödinger equations is utilized to reduce the computation of the activation rate in one-dimensional bistable potentials to variational calculation for the ground state level of a monostable quantum system. The results thus obtained are compared with the predictions of conventional approximate techniques for a class of weakly binding (soft) potentials.

In absence of structural or electronic transitions, the temperature dependence of the inelastic energy losses of escaping electrons is proved to account for most of the temperature dependence of the raw LXX Auger spectrum in vanadium. This effect is expected to be true in general.

A simple and efficient method to deduce the true Auger spectrum from the raw data is presented. This method is an application to Auger spectroscopy derived from signal processing procedures employed in different fields. It appears particularly promising when complex spectra are analyzed as it avoids spurious information to be introduced. The method is also compared with the more-standard van Cittert’s approach. An experimental check of the common assumption that an Auger electron experiences almost the same energy-loss mechanisms as a nearly elastically reflected primary electron at the same energy is given.

Aluminium L₂₃ VV and KVV Auger spectra are compared for the first time. In spite of the differences in kinetic energy of the escaping electrons and of the core-holes, the two spectra look very similar suggesting that surface effects do not contribute much to the Auger transition. Moreover in both spectra it is visible a high energy structure which is interpreted as an intrinsic plasmon-grain satellite due to the core-hole relaxation.

The equation of motion approach is employed to derive the two-body Green function, relevant for the calculation of Auger spectra, in the case of systems well described by the Hubbard Hamiltonian with many bands. The results have been used to calculate the M₁VV (M₁ valence-valence) Auger rate of copper, which has been shown to be well described by the present theory, as distinct contributions belonging to different bands are clearly present.

In questa tesi viene presentato lo studio della evoluzione temporale del tunneling risonante di elettroni attraverso microstrutture semiconduttrici. In particolare, si mostra che l’interazione elettronica è in grado di indurre un comportamento oscillante nel tempo delle correnti trasmesse e riflesse. Tali oscillazioni hanno un periodo variabile tra 10^-14 e 10^-12 secondi e la loro esistenza è suscettibile di verifica sperimentale. Il fenomeno è di rilevanza tecnologica per la progettazione di dispositivi elettronici molto piu` veloci di quelli realizzabili attualmente. Esso, inoltre, presenta un interesse concettuale in quanto costituisce un esempio di transiente quantistico con proprietà che non hanno riscontro classico. Nel capitolo 1 si introduce criticamente la fisica del trasporto quantistico in eterostrutture semiconduttrici suggerendone una interpretazione rigorosa in termini di potenziali ed equazioni efficaci allo scopo di evidenziare alcune incongruenze, quali quella di una massa efficace variabile, che potrebbero sorgere da una schema di tipo fenomenologico. Nel capitolo 2 viene discussa una equazione di Schrödinger non lineare che descrive, nel limite di campo medio, il tunneling risonante di un sistema di elettroni mutuamente interagenti. Si illustra in dettaglio l’evoluzione temporale del processo e il conseguente comportamento oscillante della carica dinamicamente intrappolata negli stati risonanti. L’approfondimento del fenomeno delle oscillazioni in relazione ad una possibile indagine sperimentale e l’analisi di alcune situazioni fisiche di particolare interesse, quali i sistemi dotati di risonanze multiple e i sistemi magnetici, costituiscono, infine, l’argomento del capitolo 3.

Come è noto uno dei problemi fondamentali della moderna indagine fisica è lo studio delle proprietà dei sistemi di particelle interagenti. In realtà è bene di stinguere tra sistemi di poche e molte particelle. Per i sistemi di poche particelle interagenti è infatti una cosa pensabile, anche se poi non fattibile, il raggiungimento di una completa conoscenza fisica del sistema in esame (cioè la conoscenza della funzione d’onda). Ma quando il numero di particelle è dell’ordine del numero di Avogadro la conoscenza della funzione d’onda del sistema diventa impossibile anche in linea di principio. Quelle che si vanno ad indagare sono allora le cosiddette proprietà termodinamiche del sistema ottenute allorché il numero delle particelle diverge mentre la loro densità rimane costante. La conoscenza esatta di queste proprietà termodinamiche rimane tuttavia un problema generalmente irrisolto e si necessita di conseguenza di tecniche di soluzione approssimata. Tra le numerose tecniche oggi a disposizione un ruolo di fondamentale importanza è rivestito dalle funzioni di Green che trovano applicazione nei più svariati campi della fisica degli stati aggregati (di bassa ed alta energia).

A questa tecnica si fa ricorso nel presente lavoro di tesi centrato sullo studio dei sistemi di molti fermioni interagenti. In particolare qui ci si riferisce ad elettroni e più precisamente alle proprietà dei metalli che possono appunto essere schematizzati, in prima approssimazione, come sistemi di elettroni mutuamente interagenti immersi in un potenziale periodico. In tale contesto fisico l’attenzione è stata focalizzata sullo studio delle cosiddette funzioni di Green a due particelle che, come verrà meglio spiegato nel seguito, descrivono il comportamento correlato di coppie di particelle. Lo studio di questa particolare classe di funzioni di Green è interessante per almeno due motivi. Innanzitutto esse permettono di ricavare le funzioni di correlazione a due particelle con la possibilità quindi di applicazioni immediate (come nel presente caso) a quei problemi caratterizzati da processi a due elettroni. Inoltre, poiché per le funzioni di Green è possibile scrivere soluzioni in termini di funzioni di ordine più alto (cioè a più particelle), la conoscenza approssimata delle funzioni di Green a due particelle porta ad un ordine di approssimazione migliore per quelle ad una particella. Troncamenti delle equazioni del moto di questo genere sono molto vantaggiosi in quanto permettono automaticamente alle funzioni ad una particella cosi trovate di soddisfare le leggi di conservazione del numero di particelle, dell’energia, della quantità e del momento della quantità di moto come indispensabilmente richiesto nella descrizione dei fenomeni di trasporto (34).

Come già accennato, nel presente studio si sfrutta solo la prima delle due possibilità offerte dalla conoscenza delle funzioni di Green, cioè la generazione delle funzioni di correlazione di coppia utili per l’investigazione delle transizioni Auger nei metalli. Per sottolineare la possibilità di ottenere ulteriori applicazioni dalle funzioni di Green qui studiate (applicazioni riguardanti sia lo studio di altri processi a due elettroni, sia lo studio di funzioni di Green ad una particella) il problema matematico della loro soluzione (capitolo 3) è diviso nettamente da quello della loro applicazione nel ricavare il rate Auger (capitolo 4). Lo studio delle funzioni di Green di interesse è condotto su 3 diversi sistemi elettronici descritti da altrettante Hamiltoniane che mettono l’accento su alcune particolari proprietà metalliche (il jellium, l’interazione elettrone-plasmone, il modello di Hubbard). Nonostante la diversità fisica dei modelli usati la tecnica delle funzioni di Green permette l’uso di un identico formalismo matematico che riduce in pratica il problema ad una ripetitiva esecuzione di commutatori canonici. Le equazioni del moto delle funzioni di Green a due particelle, semplificate nell’ambito della cosiddetta Random Phase Approximation, portano in tutti e tre i casi studiati ad un sistema di infinite equazioni lineari. Per tale sistema viene proposta una tecnica di soluzione che, mediante approssimazioni successive, porta a scrivere uno sviluppo in serie esattamente risommabile. Grazie a tale tecnica di soluzione e ad una espressione alquanto generale del rate Auger in termini di funzioni di correlazione di coppia (ricavata nel capitolo 2) è possibile trattare in modo unificato processi Auger in ambienti completamente contrastanti, quali i metalli semplici e quelli di transizione. Il confronto con i lavori sperimentali e teorici già presenti in letteratura porta a giudicare più che soddisfacenti i risultati ottenuti e suggerisce la necessità di compiere ulteriori indagini, soprattutto di natura sperimentale, per chiarire i punti ancora incerti.

Un piano più particolareggiato del presente lavoro di tesi può essere ricavato dall’indice appresso riportato. Il simbolismo e le unità di misura utilizzate, comuni nella letteratura specializzata, saranno di volta in volta illustrate secondo le esigenze.