In this thesis we will discuss systems of point interacting fermions, their stability and other spectral properties. Whereas for bosons a point interacting system is always unstable this ques- tion is more subtle for a gas of two species of fermions. In particular the answer depends on the mass ratio between these two species. Most of this work will be focused on the N + M model which consists of two species of fermions with N, M particles respectively which interact via point interactions. We will introduce this model using a formal limit and discuss the N + 1 system in more detail. In particular, we will show that for mass ratios above a critical one, which does not depend on the particle number, the N + 1 system is stable. In the context of this model we will prove rigorous versions of Tan relations which relate various quantities of the point-interacting model. By restricting the N + 1 system to a box we define a finite density model with point in- teractions. In the context of this system we will discuss the energy change when introducing a point-interacting impurity into a system of non-interacting fermions. We will see that this change in energy is bounded independently of the particle number and in particular the bound only depends on the density and the scattering length. As another special case of the N + M model we will show stability of the 2 + 2 model for mass ratios in an interval around one. Further we will investigate a different model of point interactions which was discussed before in the literature and which is, contrary to the N + M model, not given by a limiting procedure but is based on a Dirichlet form. We will show that this system behaves trivially in the thermodynamic limit, i.e. the free energy per particle is the same as the one of the non-interacting system.
We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction, and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, we identify the essential spectrum, localise and prove the finiteness of the discrete spectrum, qualify the angular symmetry of the eigenfunctions, and prove the monotonicity of the eigenvalues with respect to the mass parameter. We also demonstrate the existence of bound states in a physically relevant regime of masses.
In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ fermions of another species or in another state. There is a resonant
s-wave interaction (that is, of zero range and infinite scattering length) between fermions in different pairs and no interaction within the same pair. We study whether this
2+2 fermionic system can exhibit (as the
3+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is, the mass ratio $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, where
s is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of
s are such that
M(s) has a zero eigenvalue; here the operator
M(s), that depends on the total angular momentum $\ensuremath{\ell}$, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of
M(s), analytical and for an arbitrary imaginary
s for the continuous spectrum and numerical and limited to
s=0 and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of
M(0) crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, the continuous spectrum of
M(s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the
2+2 fermions. We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-
1/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary
s axis of
s times the logarithmic derivative of the determinant of
M(s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results.
We study analytically and numerically the binding properties, in particular the ground state, of the so-called
(2+1)-fermion system, which is a three-dimensional system of two identical fermions interacting with a third particle of different species through a zero-range interaction. We model the system with a specific self-adjoint point interaction Hamiltonian recently constructed in the mathematical literature. First we characterize the internal symmetry of the bound states in the attractive case. Then we show that the system confines in a precise regime of masses and that its ground-state energy stays finite as the mass becomes close to the critical point of the collapse of the system.