Self-adjoint extensions and spectral analysis in the Calogero problem

Abstract

In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential αx −2 . Although the problem is quite old and well-studied, we believe that our consideration, based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some "paradoxes" inherent in the "naive " quantum-mechanical treatment. We study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In addition, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented. 4 oscillator representation for the Calogero Hamiltonians.2 A "naive" treatment of the problem and related paradoxesAs mentioned above, the consideration of this section is on the so-called "physical level of rigor", or, in other words, "naive", so we actually repeat here a negative experience of the first researches. We start with the formal differential expression, or differential operation (d x = d/dx),for the Calogero Hamiltonian, and consider it as an s.a. operatorĤ in the Hilbert space H = L 2 (R + ) of quantum states for any α, conventionally without any reservations about its domain. We say in advance that the latter is precisely the reason for paradoxes. In QM, the time evolution governed by an s.a. HamiltonianĤ is unitary and is defined for all moments of time, although, as we have mentioned in Introduction, an analogue of a "fall to the center" is well-known from textbooks in the case of α < −1/4: in this case, the spectrum ofĤ is unbounded from below. This is argued [15] by considering the singular Calogero potential as a limit of bounded regularized potentials,

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

“…8 As many have observed [28][29][30][31][32][33][34][35][36][37][38][39][40] because of this the Schrödinger Hamiltonian can fail to be self-adjoint, depending the boundary conditions that hold at r = . Selecting a choice of boundary condition to secure its self-adjointness -not a unique construction -is known as constructing its self-adjoint extension [41][42][43][44][45][46][47][48].…”

“…As mentioned in [1], this boundary condition can be regarded as a specific choice of selfadjoint extension [41][42][43][44][45][46][47][48] of the inverse-square Hamiltonian. The inverse-square potential requires such an extension because its wave-functions are sufficiently bunched at the origin that physical quantities actually care about the nature of the physics encapsulated by the source action, S b .…”

Schrödinger and Dirac Operators with Aharonov–Bohm and Magnetic-Solenoid Fields

“…Its singular part was studied by us recently in [22]. We note that as in the case of the pure AB field, the division to different regions of g 1 is actually determined by the same term g 1 ρ −2 singular at the origin and independent of the value of B.…”

“…For λ a = ±π/2, they are given by the respective formulae (22) and (23) For l = l 0 , there exists a one-parameter U (1)-family of s.a. radial Hamiltoniansĥ λ (l 0 ), parameterized by the real parameter λ ∈ S (−π/2, π/2). These Hamiltonians are specified by the asymptotic s.a. boundary conditions at ρ → 0,…”

“…In constructing the Hamiltonians and performing their spectral analysis, we follow, respectively, the theory of s.a. extensions of symmetric differential operators [20,21,25] and the Krein method of guiding functionals [20,21]. Examples of similar consideration are given in [22], where a nonrelativistic particle in the Calogero and Krazter potential fields is considered, and in [23], where a Dirac particle in the Coulomb field of arbitrary charge is considered. However, due to the peculiarities of the three-dimensional (3D) problem under consideration, we use a necessary generalization of the approach [23].…”