César R. de Oliveira

Intermediate spectral theory and quantum dynamics

Preface.- Selectec Notation.- A Glance at Quantum Mechanics.- 1 Linear Operators and Spectrum.- 1.1 Bounded Operators.- 1.2 Closed Operators.- 1.3 Compact Operators.- 1.4 Hilbert-Schmidt Operators.- 1.5 Spectrum.- 1.6 Spectrum of Compact Operators.- 2 Adjoint Operator.- 2.1 Adjoint Operator.- 2.2 Cayley Transform I.- 2.3 Examples.- 2.4 Weyl Sequences.- 2.5 Cayley Transform II.- 2.6 Examples.- 3 Fourier Transform and Free Hamiltonian.- 3.1 Fourier Transform.- 3.2 Sobolev Spaces.- 3.3 Momentum Operator.- 3.4 Kinetic Energy and Free Particle.- 4 Operators via Sesquilinear Forms.- 4.1 Sesquilinear Forms.- 4.2 Operators Associated with Forms.- 4.3 Friedrichs Extension.- 4.4 Examples.- 5 Unitary Evolution Groups.- 5.1 Unitary Evolution Groups.- 5.2 Bounded Infinitesimal Generators.- 5.3 Stone Theorem.- 5.4 Examples.- 5.5 Free Quantum Dynamics.- 5.6 Trotter Product Formula.- 6 Kato-Rellich Theorem.- 6.1 Relatively Bounded Perturbations.- 6.2 Applications.- 6.3 Katos Inequality and Pointwise Positivity.- 7 Boundary Triples and Self-Adjointness.- 7.1 Boundary Forms.- 7.2 Schrodinger Operators On Intervals.- 7.3 Regular Examples.- 7.4 Singular Examples and All That.- 7.5 Spherically Symmetric Potentials.- 8 Spectral Theorem.- 8.1 Compact Self-Adjoint Operators.- 8.2 Resolution of the Identity.- 8.3 Spectral Theorem.- 8.4 Examples.- 8.5 Comments on Proofs.- 9 Applications of the Spectral Theorem.- 9.1 Quantum Interpretation of Spectral Measures.- 9.2 Proof of Theorem 5.3.1.- 9.3 Form Domain of Positive Operators.- 9.4 Polar Decomposition.- 9.5 Miscellanea.- 9.6 Spectrum Mapping.- 9.7 Duhamel Formula.- 9.8 Reducing Subspaces.- 9.9 Sequences and Evolution Groups.- 10 Convergence of Self-Adjoint Operators.- 10.1 Resolvent and Dynamical Convergences.- 10.2 Resolvent Convergence and Spectrum.- 10.3 Examples.- 10.4 Sesquilinear Forms Convergence.- 10.5 Application to the Aharonov-Bohm Effect.- 11 Spectral Decomposition I.- 11.1 Spectral Reduction.- 11.2 Discrete and Essential Spectra.- 11.3 Essential Spectrum and Compact Perturbations.- 11.4 Applications.- 11.5 Discrete Spectrum for Unbounded Potentials.- 11.6 Spectra of Self Adjoint Extensions.- 12 Spectral Decomposition II.- 12.1 Point, Absolutely and Singular Continuous Subspaces.- 12.2 Examples.- 12.3 Some Absolutely Continuous Spectra.- 12.4 Magnetic Field: Landau Levels.- 12.5 Weyl-von Neumann Theorem.- 12.6 Wonderland Theorem.- 13 Spectrum and Quantum Dynamics.- 13.1 Point Subspace: Precompact Orbits.- 13.2 Almost Periodic Trajectories.- 13.3 Quantum Return Probability.- 13.4 RAGE Theorem and Test Operators.- 13.5 Continuous Subspace: Return Probability Decay.- 13.6 Bound and Scattering States in Rn.- 13.7 alpha-Holder Spectral Measures.- 14 Some Quantum Relations.- 14.1 Hermitian x Self-Adjoint Operators.- 14.2 Uncertainty Principle.- 14.3 Commuting Observables.- 14.4 Probability Current.- 14.5 Ehrenfest Theorem.- Bibliography.- Index.

Quantum singular operator limits of thin Dirichlet tubes via Γ-convergence

The Γ -convergence of lower bounded quadratic forms is used to study the singular operator limit of thin tubes (i.e. the vanishing of the cross-section diameter) of the Laplace operator with Dinchlet boundary conditions; a procedure to obtain the effective Schrodinger operator (in different subspaces) is proposed, generalizing recent results in case of compact tubes. Finally, after scaling curvature and torsion the limit of a broken line is briefly investigated.

Self-adjoint extensions of Coulomb systems in 1, 2 and 3 dimensions

We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in R n , n = 1,2,3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.

Mathematical Justification of the Aharonov-Bohm Hamiltonian

It is presented, in the framework of nonrelativistic quantum mechanics, a justification of the usual Aharonov-Bohm hamiltonian (with solenoid of radius greater than zero). This is obtained by way of increasing sequences of finitely long solenoids together with a natural impermeability procedure; further, both limits commute. Such rigorous limits are in the strong resolvent sense and in both ℝ2 and ℝ3 spaces.

Scattering and self-adjoint extensions of the Aharonov–Bohm Hamiltonian

We consider the hamiltonian operator associated with planar sec- tions of infinitely long cylindrical solenoids and with a homogeneous magnetic field in their interior. First, in the Sobolev space

Dynamical delocalization for the 1D Bernoulli discrete Dirac operator

\mathcal H^2

Spectral and localization properties for the one-dimensional Bernoulli discrete Dirac operator

, we characterize all generalized boundary conditions on the solenoid bor- der compatible with quantum mechanics, i.e., the boundary conditions so that the corresponding hamiltonian operators are self-adjoint. Then we study and compare the scattering of the most usual boundary con- ditions, that is, Dirichlet, Neumann and Robin.

Some remarks concerning stability for nonstationary quantum systems

A 1D tight-binding version of the Dirac equation is considered; after checking that it recovers the usual discrete Schrodinger equation in the nonrelativistic limit, it is found that for two-valued Bernoulli potentials the zero-mass case presents the absence of dynamical localization for some specific values of the energy, albeit it has no continuous spectrum. For the other energy values (again excluding some very specific ones) the Bernoulli–Dirac system is localized, independently of the mass.

Ergodic hypothesis in classical statistical mechanics

An one-dimensional (1D) Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrodinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.

On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

The problem of characterizing stability and instability for general nonstationary quantum systems is investigated. Some characterizations are reported and some elementary properties of a topological characterization are established. Then, it is proven, by considering a simple example, that there are nonperiodic driven systems whose orbits are neither precompact nor leave on average any compact set. Autocorrelation measures are computed and the possible roles of the generalizes quasienergy operator and energy growth are briefly discussed.