Semiclassical analysis of systems of Schrödinger equations

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for xb. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Experimental evaluation of the z-resolution in different clinical Digital Breast Tomosynthesis systems using commercial phantoms

Digital Breast Tomosynthesis (DBT) is an advanced mammography technique based on the reconstruction of a pseudo-volumetric image. To date, image quality represents the most deficient section of DBT quality control protocols. In fact, related tests are not yet characterized by either action levels or typical values. This thesis work focuses on the evaluation of one aspect of image quality: the z-resolution. The latter is studied in terms of Artifact Spread Function (ASF), a function that describes the signal spread of a detail along the reconstructed focal planes. To quantify the ASF numerically, its Full Width at Half Maximum (FWHM) is calculated and used as a representative index of z-resolution. Experimental measurements were acquired in 24 DBT systems, of 7 different models, currently in use in 20 hospital facilities in Italy. The analysis, performed on the clinical reconstructed images, of 5 different commercial phantoms, lead to the identification of characteristic FWHM values for each type of DBT system. The ASF clearly showed a dependence on the size of the detail, providing higher FWHM values for larger objects. The z-resolution was found to be positively influenced by the acquisition angle: Fujifilm sistematically showed wider ASF profiles in ST mode (15°) than in HR mode (40°). However, no clear relationship was found between angular range and ASF, among different DBT systems, due to the influence of the peculiarities of each reconstruction algorithm. The experimental approach shown in this thesis work can be proposed as a z-resolution quality control test procedure. Contextually, the values found could be used as a starting point for identifying typical values to be included in the test, in a DBT protocol. Clearly, a statistically significant number of images is needed to do this. The equipment involved in this work is located in hospitals and is not available for research purposes, so only a limited amount of data was acquired and processed.