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We prove that in two dimensions the contact interaction of two wave functions is represented by a self-adjoint operator. Together with a confining potential, this provides a stable condensate of two-particle systems. Recall that in three dimensions one has a condensate of four wave functions in contact interaction (the Bose–Einstein condensate).

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass m, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for m larger than a critical value m∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H0 can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for m ∈ (m∗,m∗∗), where m∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H0,β, β ∈ ℝ, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.

It is an experimental fact that (Formula presented.) -decays produce in a cloud chamber at most one track (sequence of liquid droplets) and that this track points in a random direction. This seems to contradict the description of decay in Quantum Mechanics: according to Gamow a spherical wave is produced and moves radially according to Schrödinger’s equation. It is as if the interaction with the supersaturated vapor turned the wave into a particle. The aim of this note is to place this effect in the context of Schrödinger’s Quantum Mechanics. We shall see that the properties of the initial wave function suggest the introduction of a semiclassical formalism in which the (Formula presented.) -wave can be described as a collection of semiclassical (probability) wavelets; each of them interacts with an atom and forms an entangled state. The interaction can be regarded as a semiclassical inelastic scattering event. The measurement (of the position of the first droplet of the track) selects the wave function of one of the ions, with a probability given by Born’s rule. This ion interacts with the atoms nearby leading to the formation of a droplet. One can reasonably assume that also the wavelet entangled with the selected ion has probability one to remain as part of the description of the system. The measurement process is therefore represented by a (non local) unitary operator. The (semiclassical) wavelet remains sharply localized on a classical path (Formula presented.) It is still a probability wave: it determines the probability that another atom be ionized. This probability is essentially zero unless the atom lies on (Formula presented.). This gives the visible track.

The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula. The second part (Selected Topics) are lecture notes of a more advanced course aimed at giving the basic notions necessary to do research in several areas of mathematical physics connected with quantum mechanics, from solid state to singular interactions, many body theory, semi-classical analysis, quantum statistical mechanics. The structure of this book is suitable for a second-semester course, in which the lectures are meant to provide, in addition to theorems and proofs, an overview of a more specific subject and hints to the direction of research. In this respect and for the width of subjects this second volume differs from other monographs on Quantum Mechanics. The second volume can be useful for students who want to have a basic preparation for doing research and for instructors who may want to use it as a basis for the presentation of selected topics.