Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians

The study of a quantum particle on degenerate Riemannian manifolds, and the problem of the purely geometric confinement away from the singularity locus of the metric, as opposite to the dynamical transmission across the singularity, has recently attracted a considerable amount of attention in relation to Grushin structures and to the induced confining effective potentials on cylinder, cone, and plane, as well as, more generally, on two-step two-dimensional almost-Riemannian structures, or also generalisations to almost-Riemannian structures in any dimension and of any step, and even to sub-Riemannian geometries, provided that certain geometrical assumptions on the singular set are taken. On a related note, a satisfactory interpretation of the heat-confinement in the Grushin cylinder is known in terms of Brownian motions. Underlying such analyses there is a natural problem of control of essential self-adjointness or lack thereof, whence also a natural problem of identification, classification, and analysis of self-adjoint extensions, for the minimally defined Laplace-Beltrami operator on manifold. Such questions are discussed in this Chapter for the planar, and, more extensively, the cylindric realisation of a Grushin-type structure, where both the self-adjointness characterisation (Sects. 5.4–5.12) and the spectral analysis (Sects. 5.13 and 5.14) are carried on by means of the Kreı̆n-Višik-Birman extension scheme. The whole chapter is modelled on the recent works (Gallone et al., Z. Angew. Math. Phys. 70, Art. 158, 17, 2019; E. Pozzoli, Quantum confinement in α-Grushin planes, in Mathematical Challenges of Zero-Range Physics, ed. by A. Michelangeli. Springer INdAM Series (Springer, Berlin, 2021), pp. 229–237;M. Gallone, A. Michelangeli, E. Pozzoli, Quantum geometric confinement and dynamical transmission in Grushin cylinder (2020). arXiv:2003.07128; M. Gallone, A. Michelangeli, J. Phys. A Math. Theor. 2021; M. Gallone, A. Michelangeli, E. Pozzoli, Heat equation with inverse-square potential of bridging type across two half-lines (2021). arXiv:2112.01255).

Models of particles coupled by a non-trivial interaction whose spatial range is zero represent one of the deepest modern applications in quantum mechanics of self-adjoint extension theory. This is also a field that combines a long history of well-established results with really hard problems that are still open. This is even more so in view of the rather intriguing circumstance that recent experimental advances have made the subject highly topical in theoretical and experimental cold atom physics, in parallel with the purely mathematical investigation, a scenario where formal physical heuristics produce beautiful results and conjectures that still need be demonstrated in full mathematical rigour. Self-adjoint extension theory kicks in to identify physically meaningful self-adjoint realisations of formal Hamiltonians, and then to study their spectral properties. Mathematically this is often particularly challenging because in this context one is not dealing with ordinary Schrödinger operators with usual structure of kinetic plus potential part: indeed, the ‘interaction potential’ is rather to be thought of as some sort of delta-like profile (with the important warning, as specified later, that in two and three dimensions these are not and cannot be Dirac deltas), thus a large amount of techniques that are specific from Schrödinger operator theory are not directly applicable. While giving multiple references to a growing and active literature, this chapter is focussed on a specific class of zero-range models, in fact on one of the most relevant and instructive ‘building blocks’, namely a system of three identical bosons with non-trivial contact interaction, keeping the recent work (A. Michelangeli, Rev. Math. Phys. 33, 2150010, 101, 2021) as main reference. There is also a pedagogical motivation for choosing such bosonic playground: as these models are naturally unbounded from below, it turns out that a clever combination of both von Neumann and Kreı̆n-Višik-Birman extension schemes is required.