Time delay for one-dimensional quantum systems with steplike potentials

This paper concerns time-dependent scattering theory and in particular the concept of time delay for a class of one-dimensional anisotropic quantum systems. These systems are described by a Schr\"{o}dinger Hamiltonian $H = -\Delta + V$ with a potential $V(x)$ converging to different limits $V_{\ell}$ and $V_{r}$ as $x \to -\infty$ and $x \to +\infty$ respectively. Due to the anisotropy they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where $V(x)$ vanishes as $|x| \to \infty$ ($V_\ell = V_r$). We show that it remains true in the anisotropic case $V_\ell \not = V_r$, i.e. we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression which is commonly used for time delay in the literature.Comment: 48 pages, 1 figur

On the Localization of One-Photon States

Single photon states with arbitrarily fast asymptotic power-law fall-off of energy density and photodetection rate are explicitly constructed. This goes beyond the recently discovered tenth power-law of the Hellwarth-Nouchi photon which itself superseded the long-standing seventh power-law of the Amrein photon.Comment: 7 pages, tex, no figure

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 $\mathcal H^2$, 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.Comment: 40 pages, 5 figure

Rigorous Real-Time Feynman Path Integral for Vector Potentials

we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the $L^2$ transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities

The EPATH trial

Observational studies suggested a link between bone disease and left ventricular (LV) dysfunction that may be pronounced in hyperparathyroid conditions. We therefore aimed to test the hypothesis that circulating markers of bone turnover correlate with LV function in a cohort of patients with primary hyperparathyroidism (pHPT). Cross-sectional data of 155 subjects with pHPT were analyzed who participated in the “Eplerenone in Primary Hyperparathyroidism” (EPATH) Trial. Multivariate linear regression analyses with LV ejection fraction (LVEF, systolic function) or peak early transmitral filling velocity (e’, diastolic function) as dependent variables and N-terminal propeptide of procollagen type 1 (P1NP), osteocalcin (OC), bone- specific alkaline phosphatase (BALP), or beta-crosslaps (CTX) as the respective independent variable were performed. Analyses were additionally adjusted for plasma parathyroid hormone, plasma calcium, age, sex, HbA1c, body mass index, mean 24-hours systolic blood pressure, smoking status, estimated glomerular filtration rate, antihypertensive treatment, osteoporosis treatment, 25-hydroxy vitamin D and N-terminal pro-brain B-type natriuretic peptide. Independent relationships were observed between P1NP and LVEF (adjusted β-coefficient = 0.201, P = 0.035) and e’ (β = 0.188, P = 0.042), respectively. OC (β = 0.192, P = 0.039) and BALP (β = 0.198, P = 0.030) were each independently related with e’. CTX showed no correlations with LVEF or e’. In conclusion, high bone formation markers were independently and paradoxically related with better LV diastolic and, partly, better systolic function, in the setting of pHPT. Potentially cardio-protective properties of stimulated bone formation in the context of hyperparathyroidism should be explored in future studies

The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part I

We exemplify the way the rigged Hilbert space deals with the Lippmann-Schwinger equation by way of the spherical shell potential. We explicitly construct the Lippmann-Schwinger bras and kets along with their energy representation, their time evolution and the rigged Hilbert spaces to which they belong. It will be concluded that the natural setting for the solutions of the Lippmann-Schwinger equation--and therefore for scattering theory--is the rigged Hilbert space rather than just the Hilbert space.Comment: 34 pages, 1 figur

The various power decays of the survival probability at long times for free quantum particle

The long time behaviour of the survival probability of initial state and its dependence on the initial states are considered, for the one dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state $\psi (x)$, which is assumed to decrease sufficiently rapidly at large $|x|$. We then show that the behaviour of the survival probability at long times is determined by that of the initial state $\psi$ at zero momentum $k=0$. Indeed, it is proved that the survival probability can exhibit the various power-decays like $t^{-2m-1}$ for an arbitrary non-negative integers $m$ as $t \to \infty$, corresponding to the initial states with the condition $\hat{\psi} (k) = O(k^m)$ as $k\to 0$.Comment: 15 pages, to appear in J. Phys.