arXiv subjects
Publications and source records attributed to Vincenzo Grecchi.
We consider a family of quantum Hamiltonians $H_\hbar=-\hbar^2\,(d^2\!/dx^2) +V(x)$, $x\in\mathbb{R},$ $\hbar>0,$ where $V(x)=i(x^3-x)$ is an imaginary double well potential. We prove the existence of infinite eigenvalue crossings with the selection rules of the eigenvalue pairs taking part in a crossing. This is a semiclassical localization effect. The eigenvalues at the crossings accumulate at a critical energy for some of the Stokes lines.
We consider a \textit{PT}-symmetric cubic oscillator with an imaginary double well. We prove the existence of an infinite number of level crossings with a definite selection rule. Decreasing the positive parameter $\hbar$ from large values, at a value $\hbar_n$ we find the crossing of the pair of levels $(E_{2n+1}(\hbar),E_{2n}(\hbar))$ becoming the pair of levels $(E_n^+(\hbar),E_n^-(\hbar))$. For large parameters, a level is a holomorphic function $E_m(\hbar)$ with different semiclassical behaviors, $E_j^\pm(\hbar),$ along different paths. The corresponding $m$-nodes delocalized state $ψ_m(\hbar)$ behaves along the same paths as the semiclassical $j$-nodes states $ψ_j^\pm(\hbar),$ localized at one of the wells $x_\pm$ respectively. In particular, if the crossing parameter $\hbar_n$ is by-passed from above, the levels $E_{2n+(1/2)\pm(1/2)}(\hbar)$ have respectively the semiclassical behaviors of the levels $E_n^\mp(\hbar)$ along the real axis. These results are obtained by the control of the nodes. There is evidence that the parameters $\hbar_n$ accumulate at zero and the accumulation point of the corresponding energies is aninstability point of a subset of the Stokes complex called the monochord, consisting of the vibrating string and the sound board.
We make a nodal analysis of the processes of level crossings in a model of quantum mechanics with a PT-symmetric double well. We prove the existence of infinite crossings with their selection rules. At the crossing, before the PT-symmetry breaking and the localization, we have a total P-symmetry breaking of the states.
We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$H(β)=-d^2/dx^2+x^2+i\sqrtβx^3$,for $β$ in the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $\{E_n(β)\}_{n\geq 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $β\in\C_c$, $E_n(β)$ can be computed as the Stieltjes-Padé sum of its perturbation series at $β=0$. This also gives an alternative proof of the fact that the spectrum of $H(β)$ is real when $β$ is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.
We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the width. Similar results are found for the Klein-Gordon oscillators, and in this case there is an infinite number of dynamics and the eigenvalues and eigenvectors of the PT-symmetric operators give metastable states for each dynamics.
We consider the vibrational energy levels of the first two electronic states of the molecule ion $H_2^+$. The Born-Oppenheimer method applied to the case of the Stark effect on a $H_2^+$-like molecule gives existence of sharp resonances localized in the same interval of energy of the vibrational levels.
In this paper we review some known results on the motion of Bloch Oscillators in the crystal momentum representation. We emphasize that the acceleration theorem, as usually stated by most of the authors, is incomplete, but in the case of pure Bloch states. In fact, the exact version of the acceleration theorem should contain a phase factor depending on both the time and crystal momentum variables. As we show this phase factor plays an essential role in order to understand the motion of Bloch Oscillators in the position variable.
Here, we show how the molecular structure appears and becomes stable for supercritical physical conditions. In particular we consider, for the ammonia molecule in a gas, a simple model based on a standard non-linear double-well Schroedinger equation with a dissipative term and a term representing weak collisions.