arXiv subjects
Publications and source records attributed to Sandro Graffi.
It is established that a PT-symmetric elliptic quadratic differential operator with real spectrum is similar to a self-adjoint operator precisely when the associated fundamental matrix has no Jordan blocks.
A class of non-selfadjoint, $\PT$-symmetric operators is identified similar to a self-adjoint one, thus entailing the reality of the spectrum. The similarity transformation is explicitly constructed through the method of the quantum normal form, whose convergence (uniform with respect to the Planck constant) is proved. Further consequences of the uniform convergence of the quantum normal form are the establishment of an exact quantization formula for the eigenvalues.
We consider the Schrödinger operator defined by the quantization of the linear flow of diophantine frequencies over the l-dimensional torus, perturbed by a holomorphic potential which depends on the actions only through their particular linear combination defining the Hamiltonian of the linear flow. We prove that the corresponding quantum normal form converges uniformly with respect to the Planck constant. This result simultaneously yields an exact quantization formula for the quantum spectrum, as well as a convergence criterion for the Birkhoff normal form, valid for a class of perturbations holomorphic away from the origin.
We construct a family of Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding Hamilton-Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.
We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^ε$ for $-1<ε<0$.
The evolution of a quantum system subjected to infinitely many measurements in a finite time interval is confined in a proper subspace of the Hilbert space. This phenomenon is called "quantum Zeno effect": a particle under intensive observation does not evolve. This effect is at variance with the classical evolution, which obviously is not affected by any observations. By a semiclassical analysis we will show that the quantum Zeno effect vanishes at all orders, when the Planck constant tends to zero, and thus it is a purely quantum phenomenon without classical analog, at the same level of tunneling.
The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.
Consider in $L^2(\R^2)$ the operator family $H(ε):=P_0(\hbar,ω)+εF_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\om$, $F_0$ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and $\ep\in\C$. Then there exist $\ep^\ast >0$ independent of $\hbar$ and an open set $Ω\subset\C^2\setminus\R^2$ such that if $|\ep|<\ep^\ast$ and $\om\in\Om$ the quantum normal form near $P_0$ converges uniformly with respect to $\hbar$. This yields an exact quantization formula for the eigenvalues, and for $\hbar=0$ the classical Cherry theorem on convergence of Birkhoff's normal form for complex frequencies is recovered.
We present a new proof of the convergence of the N-particle Schroedinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in the Planck constant , up to an exponentially small remainder. For h=0, the classical dynamics in the mean-field limit is given by the Vlasov equation.
Criteria are formulated both for the existence and for the non-existence of complex eigenvalues for a class of non self-adjoint operators in Hilbert space invarariant under a particular discrete symmetry. Applications to the PT-symmetric Schrödinger operators are discussed.
For the Edwards-Anderson model we introduce an integral representation for the surface pressure (per unit surface) in terms of a quenched moment of the bond-overlap on the surface. We find upper and lower bounds uniformly in the volume and show that at high temperature its value is strictly different from zero.
Consider the non-compact billiard in the first quandrant bounded by the positive $x$-semiaxis, the positive $y$-semiaxis and the graph of $f(x) = (x+1)^{-α}$, $α\in (1,2]$. Although the Schnirelman Theorem holds, the quantum average of the position $x$ is finite on any eigenstate, while classical ergodicity entails that the classical time average of $x$ is unbounded.
Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity condition on the matrix elements.
If the variance of a short range Gaussian random potential grows like the volume its quenched thermodynamic limit is reached monotonically.
We prove that, for a smooth two-body potentials, the quantum mean-field approximation to the nonlinear Schroedinger equation of the Hartree type is stable at the classical limit h \to 0, yielding the classical Vlasov equation.
For the long-range deterministic spin models with glassy behaviour of Marinari, Parisi and Ritort we prove weighted factorization properties of the correlation functions which represent the natural generalization of the factorization rules valid for the Curie-Weiss case.
We eliminate by KAM methods the time dependence in a class of linear differential equations in $\ell^2$ subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator $ H_0+εP(\om t)$ for $ε$ small. Here $H_0$ is the one-dimensional Schrödinger operator $p^2+V$, $V(x)\sim |x|^α, α>2$ for $|x|\to\infty$, the time quasi--periodic perturbation $P$ may grow as $\displaystyle |x|^β, β<(α-2)/{2}$, and the frequency vector $ω$ is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non constant coefficients.
We prove that the Floquet spectrum of a class of time-periodic Schroedinger equations under a a mildly nonlinear resonant forcing is purely absolutely continuous.