DeWitt’s covariant formulation of path integration [B. De Witt, “Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,” Rev. Mod. Phys. 29, 377–397 (1957)] has two practical advantages over the traditional methods of “lattice approximations;” there is no ordering problem, and classical symmetries are manifestly preserved at the quantum level. Applying the spectral theorem for unbounded self-adjoint operators, we provide a rigorous proof of the convergence of certain path integrals on Riemann surfaces of constant curvature −1. The Pauli–DeWitt curvature correction term arises, as in DeWitt’s work. Introducing a Fuchsian group Γ of the first kind, and a continuous, bounded, Γ-automorphic potential V, we obtain a Feynman–Kac formula for the automorphic Schrödinger equation on the Riemann surface Γ\ℍ. We analyze the Wick rotation and prove the strong convergence of the so-called Feynman maps [K. D. Elworthy, Path Integration on Manifolds, Mathematical Aspects of Superspace, edited by Seifert, Clarke, and Rosenblum (Reidel, Boston, 1983), pp. 47–90] on a dense set of states. Finally, we give a new proof of some results in C. Grosche and F. Steiner, “The path integral on the Poincare upper half plane and for Liouville quantum mechanics,” Phys. Lett. A 123, 319–328 (1987).
In this dissertation, we develop and apply a relativistic framework for studying three-body scattering amplitudes, their analytic structure, and the emergence of universal phenomena such as the Efimov effect. The work integrates three complementary components. First, we introduce a systematically improvable numerical method for solving relativistic three-body integral equations in momentum space. By discretizing the continuum problem into finite matrix equations and extrapolating to the continuum limit, we obtain stable partial-wave amplitudes in the presence of a two-body bound state. Two complementary treatments of the pole contribution are implemented and shown to reproduce previous finite volume results, with controlled estimates of systematic uncertainties. Second, we provide a general prescription for analytically continuing these integral equations into the complex energy plane. Using contour-deformation techniques guided by the singularity structure of the kernel, we compute amplitudes below elastic thresholds and on multiple Riemann sheets. This allows us to determine three-body bound and virtual states consistent with earlier finite-volume studies, and to identify departures from the standard two-body finite-volume formalism due to left-hand cuts. Finally, we apply this framework to study Efimov physics in a fully relativistic setting. By tuning the interaction toward unitarity, we observe the characteristic discrete scaling symmetry of Efimov trimers and recover the expected geometric energy pattern. Tracking trimer poles across physical and unphysical Riemann sheets, we reveal rich trajectories connecting virtual states, bound states, and resonances. We also uncover cyclic behavior of higher Efimov resonances and propose a partial resolution to the associated “missing states” puzzle in terms of the multi-sheeted structure induced by the logarithmic three-body unitarity cut. Overall, this work establishes a coherent relativistic approach to the three-body problem, clarifies its analytic structure, and demonstrates its ability to capture universal features relevant to nuclear, hadronic, and atomic few-body systems.
The potential $V(x)=\ensuremath{-}{x}^{4}$, which is unbounded below on the real line, can give rise to a well-posed bound state problem when
x is taken on a contour in the lower-half complex plane. It is then
PT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integrable systems, was subsequently constructed for the general class of potentials $\ensuremath{-}(ix{)}^{N}$. For such Hamiltonians the natural
PT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator
Q. Further, with the help of this operator an equivalent Hermitian Hamiltonian
h can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential
m2x2+igx3. However, until now, the $\ensuremath{-}{x}^{4}$ potential has proved intractable. In the present paper we give explicit, closed form expressions for
Q and
h, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.