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We show that it is possible to associate univocally with each given solution of the time-dependent Schrödinger equation a particular phase flow (“quantum flow”) of a non-autonomous dynamical system. This fact allows us to introduce a definition of chaos in quantum dynamics (quantum chaos), which is based on the classical theory of chaos in dynamical systems. In such a way we can introduce quantities which may be appelled quantum Lyapunov exponents. Our approach applies to a non-relativistic quantum-mechanical system of n charged particles; in the present work numerical calculations are performed only for the hydrogen atom. In the computation of the trajectories we first neglect the spin contribution to chaos, then we consider the spin effects in quantum chaos. We show how the quantum Lyapunov exponents can be evaluated and give several numerical results which describe some properties found in the present approach. Although the system is very simple and the classical counterpart is regular, the most non-stationary solutions of the corresponding Schrödinger equation are chaotic according to our definition.

The gradient (steepest descent) method is developed in a suitable topology for the calculation of isolated eigenvalues and eigenvectors of a self-adjoint, semi-bounded operator in a Hilbert space. Sufficient conditions for the convergence of successive approximations for the eigenvalues and eigenvectors are given. The method can be applied to a wide class of Hamiltonian operators and appears to be advantageous compared with perturbation theory, both from the mathematical and the practical point of view. Si sviluppa in un'opportuna topologia il metodo del gradiente (della più ripida discesa) per il calcolo di autovalori discreti ed autovettori di un operatore autoaggiunto e semilimitato in uno spazio di Hilbert. Si danno delle semplici condizioni sufficienti per la convergenza delle approssimazioni successive agli autovalori ed agli autovettori. Il metodo può essere applicato ad un'ampia classe di operatori Hamiltoniani e sembra vantaggioso in paragone alla teoria delle perturbazioni sia da un punto di vista matematico, sia da uno pratico. Развивается градиентный метод (наибыстрейшего спуска) в соответствующей топологии для вычисления изолированных собственных значений и собственных векторов самосопряженного, полусвязанного оператора в Гильбертовом пространстве. Приводятся достаточные условия для сходимости последовательных приближений для собственных значений и собственных векторов. Предложенный метод может быть использован для широкого класса операторов Гамильтона и обладает преимуществами по сравнению с теорией возмущений с математической и практической точек зрения.

A rigorous theory of variational principles in quantum mechanics is formulated. This result is obtained by generalizing the standard theory of the inverse problem of the variational caleulus, according to new concepts of differentiability on complex Banach spaces. Our theory provides sound mathematical foundations to all known variational principles of quantum mechanics and a powerful tool to find new ones. As a simple application, we derive rigorously two variational principles for the time-dependent Schrödinger equation: the conventional one, requiring both an initial and a final condition, and a new one, requiring an initial condition only.

We obtain, by completing the differential calculus on complex Banach spaces, introduced by Sharma and Rebelo: 1) the rigorous mathematical framework of quantum-mechanical variational problems; 2) a family of iterative minimization methods, convergent, under suitable conditions, to eigenvalues and eigenvectors of the discrete spectrum of Hamiltonian operators. Applications of these methods to atomic systems are given and finally some of their advantages are discussed.

An antiserum against cabergoline, a powerful dopamine-agonist under clinical trials for the treatment of Parkinson's disease and hyperprolactinemia, has been raised in rabbits by immunization with an immunogen produced by conjugation of cabergoline to bovine serum albumin.The antiserum was able to bind a derivative of cabergoline labelled with tritium and was able to distinguish the drug molecule from some of its close related compounds and from other agents that could be simultaneously present in plasma from patients undergoing treatment with cabergoline.The antiserum and the tritium labelled hapten were used to develop a radioimmunoassay for cabergoline determination in human plasma and urine.A linear relationship between cabergoline added and % radioactivity bound was found in the range 1.9-500 pg/tube.The addition in the assay of 200 μL human plasma or 25 μL urine did not affect the specific and the non-specific binding of the radiolabelled hapten so enabling us to obtain a final sensitivity of about 12 pg/ml plasma and 120 pg/ml urine. The assay was validated in terms of reproducibility, precision and accuracy over the whole range of concentrations tested both in plasma and urine.The plasma concentrations at the steady state in a patient with Parkinson's disease who had received the drug at single oral daily doses of 3, 5 and 7 mg were determined using the assay.

We present the gradient method as a numerical procedure for the discrete spectrum of Hamiltonian operators in a version which is more general and has better numerical flexibility than a previous version. We show that the gradient method, in addition to have relationships to the Rayleigh-Schrödinger perturbation theory and to the variational method, has also some advantageous features in comparison with them and even in comparison with the Padé and Borel summability methods. In order to check the numerical efficiency of our procedure, we have applied it, with encouraging results, to the calculation of the ground-state energy of the anharmonic oscillatorsp2+x2+βx4andp2+x2+βx8for a large range of β.

We suggest to use the Newton iteration method for constructing a (locally unique) solution of the atomic and nuclear Hartree-Fock equations for an arbitrary number of particles. Our proposal is based on a theorem by Kantorovic and rests on the following points: 1) the two-body potential must satisfy a boundedness condition; 2) the zero-order approximation, used to start the iteration sequence, must satisfy certain conditions, to be proved numerically. Condition 1) holds, for instance, for all local potentials, defined by a bounded function and for a class of nonlocal potentials; it does not hold for local potentials, behaving as 1/r near the origin.

The gradient method, introduced in previous papers as a new method for solving Schrödinger equations, is applied here to compute the first two eigen values and the corresponding eigenfunctions of the Hamiltonians H(β) =p2+x2+ βx2q, withq= 2, 3, 4 and 10-4≤ β ≤ 104. The results, which for smallqare excellent for all β, are compared with those obtained by other authors.

The necessary and sufficient conditions for an abstract equation in a complex Banach space to admit a variational principle, which one of us described in a previous paper, are here applied to a system of nonlinear partial differential equations of arbitrary order in a complexL2space. More specific and helpful conditions are thus obtained. General consequences and special implications for some classes of nonlinear Schröndinger equations are also discussed and differences from an analogous result valid only in real spaces are put in evidence.

We present a rigorous theory of variational principles for equationsA(x)=0,x∈D, whereDis a suitable domain andAan operator from one complex Banach space to another, differentiable according to a differential calculus previously introduced. The theory can be applied in every branch of theoretical physics, but it has specifically been conceived to solve the peculiar difficulties, which the problem of a rigorous formulation of variational principles in quantum mechanics presents.