arXiv subjects

O. Teoman Turgut

Publications and source records attributed to O. Teoman Turgut.

At least 19 recordsLinked to original sources

arXiv2026

Perturbation Theory for Time-Dependent Point Interactions with Discrete Spectrum

We develop a perturbative framework for time-dependent point interactions in quantum systems with purely discrete unperturbed spectrum. In two and three dimensions the point interaction is described through the renormalized resolvent obtained from heat-kernel regularization, while in one dimension the diagonal Green function is finite and no renormalization is required. Time dependence is introduced either through the interaction parameter or through the motion of the support point. In both cases we expand the non-autonomous Hamiltonian in the spectral basis of the corresponding static point-interaction problem and derive the first- and second-order pole shifts, projection corrections, and transition amplitudes. We also describe how finite degeneracies are treated by separating the coupled point-interaction direction from the unaffected orthogonal subspace in each degenerate eigenspace. The method is illustrated by explicit examples: point interactions with time-dependent coupling, harmonic oscillators perturbed by moving point interactions, and a particle on a sphere with a moving interaction center. We also discuss the one-dimensional harmonic oscillator with a moving delta potential, showing that in the non-renormalized case the present formulation reduces to the standard time-dependent perturbation theory. Finally, an exactly solvable delta center moving with constant velocity on a circle is used as a discrete-spectrum example for the local moving-center expansion and its velocity-dependent dynamical corrections.

math-ph↗

arXiv2026

BCS Gap Equation on Riemannian Manifolds: A Heat Kernel Analysis

In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit ($Δ\ll μ$). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature ($\partial Δ/ \partial T < 0$), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.

hep-th↗

arXiv2026

Euler-Poincare Reduction: A Geometric Treatment of the Relativistic Spinning Particle

The heavy top is a classic example of Euler-Poincare reduction on a semidirect product group, demonstrating how an advected quantity-the direction of gravity-couples to rotational motion. While the description of relativistic spinning particles via Lie group configurations is well established in Hamiltonian (symplectic) mechanics, we demonstrate that an explicit Lagrangian Euler-Poincare reduction provides a direct, transparent parallel to the heavy top. Specifically, we show that the free relativistic spinning particle on the Poincare group is the precise Lorentzian analog of this system, with spacetime position being the advected quantity. This geometric unification not only illuminates the underlying symmetry of both systems but also clarifies why the introduction of external electromagnetic fields leads to a partial reduction. Finally, we establish that the transition to left-trivialized coordinates acts as a geometric selection mechanism, naturally favoring a class of interaction Lagrangians where spin-field coupling manifests as a dynamical redefinition of the particle's mass.

math-ph↗

arXiv2026

Renormalization of One-Dimensional Semirelativistic Bosons with Contact Interactions

We study one-dimensional spinless bosons with semirelativistic spinless-Salpeter dispersion and attractive pairwise contact interactions. Because the dispersion becomes linear at large momentum, the contact interaction is marginal by power counting and produces a logarithmic ultraviolet divergence. We construct the renormalized many-body theory using an enlarged Fock space and a Schur-complement representation of the resolvent, eliminating the bare coupling in favor of the physical zero-total-momentum two-body bound-state energy. The resulting cutoff-independent resolvent defines a self-adjoint Hamiltonian in each fixed particle-number sector. We treat the two-body problem explicitly and show, in the norm-resolvent sense, that the nonrelativistic limit reproduces the attractive Lieb--Liniger Hamiltonian. We also formulate a mean-field approximation directly within the renormalized theory. In the massless and deeply bound large-particle-number regimes, it predicts an exponentially increasing binding scale whose exponent is determined by a one-dimensional variational problem.

cond-mat.quant-gas↗

arXiv2026

On the Thermodynamic Limit of Bogoluibov's Theory of Bose Gas

Assuming that Bogoliubov's theory of weakly interacting dilute Bose gas defines a self-consistent model Hamiltonian, we investigate its thermodynamic limit as we take the volume to infinity. The infinite volume is taken via a sequence of scaled convex regions with piecewise smooth boundary and the volumes staying proportional to the cube of the diameter of the region. To get a strict bound on the behavior of the thermodynamic limit, we use the recent formulation of Bogoliubov's theory of condensation in terms of heat kernels for a given domain as well as an estimate of the difference of traces between the heat kernel with Neumann boundary conditions on this domain and the infinite space result. We cannot control the limiting process by the area term; however, we can come arbitrarily close to it.

math-ph↗

arXiv2025

Completeness Relation in Renormalized Quantum Systems

In this work, we show that the completeness relation for the eigenvectors, which is an essential assumption of quantum mechanics, remains true if the Hamiltonian, having a discrete spectrum, is modified by a delta potential (to be made precise by a renormalization scheme) supported at a point in two and three-dimensional compact manifolds or Euclidean spaces. The formulation can be easily extended to $N$ center case, and the case where delta interaction is supported on curves in the plane or space. We finally give an interesting application for sudden perturbation of the support of the delta potential.

quant-ph↗

arXiv2023

On Schrödinger Operators Modified by $δ$ Interactions

We study the spectral properties of a Schrödinger operator $H_0$ modified by $δ$ interactions and show explicitly how the poles of the new Green's function are rearranged relative to the poles of original Green's function of $H_0$. We prove that the new bound state energies are interlaced between the old ones, and the ground state energy is always lowered if the $δ$ interaction is attractive. We also derive an alternative perturbative method of finding the bound state energies and wave functions under the assumption of a small coupling constant in a somewhat heuristic manner. We further show that these results can be extended to cases in which a renormalization process is required. We consider the possible extensions of our results to the multi center case, to $δ$ interaction supported on curves, and to the case, where the particle is moving in a compact two-dimensional manifold under the influence of $δ$ interaction. Finally, the semi-relativistic extension of the last problem has been studied explicitly.

math-ph↗

arXiv2023

Relativistic Lee Model and its Resolvent Analysis

We reexamine the relativistic 2+1 dimensional Lee model in light-front coordinates on flat space and on a space-time with a spatial section given by a compact manifold in the usual canonical formalism. The simpler 2+1 dimension is chosen because renormalization is needed only for the mass difference but not required for the coupling constant and the wavefunction. The model is constructed non-perturbatively based on the resolvent formulation [1]. The bound state spectrum is studied through its ``principal operator" and bounds for the ground state energy are obtained. We show that the formal expression found indeed defines the resolvent of a self-adjoint operator--the Hamiltonian of the interacting system. Moreover, we prove an essential result that the principal operator corresponds to a self-adjoint holomorphic family of type-A in the sense of Kato.

math-ph↗

arXiv2022

Generating Axially Symmetric Minimal Hyper-surfaces in R^{1,3}

It is shown that, somewhat similar to the case of classical Baecklund transformations for surfaces of constant negative curvature, infinitely many axially symmetric minimal hypersurfaces in 4-dimensional Minkowski-space can be obtained, in a non-trivial way, from any given one by combining the scaling symmetries of the equations in light cone coordinates with a non-obvious symmetry (the analogue of Bianchis original transformation) - which can be shown to be involutive/correspond to a space-reflection.

math.DG↗

arXiv2022

Comment on "Scattering of light by a parity-time-symmetric dipole beyond the first Born approximation"

In [J. A. Rebouças and P. A. Brandão, Phys. Rev. A 104, 063514 (2021)] the authors compute the scattering amplitude for a $\mathcal{P}\mathcal{T}$-symmetric double-delta-function potential in three dimensions by invoking the far-zone approximation and summing the resulting Born series. We show that the analysis of this paper suffers from a basic error. Therefore its results are inconclusive. We give an exact closed-form expression for the scattering amplitude of this potential.

quant-ph↗

arXiv2019

A Perturbative Approach to the Tunneling Phenomena

The double-well potential is a good example, where we can compute the splitting in the bound state energy of the system due to the tunneling effect with various methods, namely WKB or instanton calculations. All these methods are non-perturbative and there is a common belief that it is difficult to find the splitting in the energy due to the barrier penetration from a perturbative analysis. However, we will illustrate by explicit examples containing singular potentials (e.g., Dirac delta potentials supported by points and curves and their relativistic extensions)that it is possible to find the splitting in the bound state energies by developing some kind of perturbation method.

quant-ph↗

arXiv2019

An attractive $ϕ^4$ theory in light-front coordinates

We study an attractive $ϕ^4$ interaction using Tamm-Dancoff truncation with light-front coordinates in $3+1$ dimensions. The truncated theory requires a coupling constant renormalization, we compute its $β$ function non-perturbatively, show that the model is asymptotically free, and find the corresponding Callan-Symanzik equations. The model supports bound states, we find the wave function for the ground state of the two-particle sector. We also give a bound for the $N$-particle ground state energy within a mean field approximation, including the corresponding result for the case of $2+1$ dimensions where the model does not require renormalization.

hep-th↗

arXiv2017

New Minimal Hypersurfaces in R(k+1)(2k+1) and S(2k+3)k

We find a class of minimal hypersurfaces H(k) as the zero level set of Pfaffians, resp. determinants of real 2k+2 dimensional antisymmetric matrices. While H(1) and H(2) are congruent to a 6-dimensional quadratic cone resp. Hsiang's cubic su(4) invariant in R15, H(k>2) (special harmonic so(2k+2)-invariant cones of degree>3) seem to be new.

math.DG↗

arXiv2016

Born-Oppenheimer Approximation for a Simple Renormalizable System

We discuss a simple singular system in two dimension, two heavy particles interacting with a light particle via an attractive contact interaction. Although intuitively clear the actual application of the Born-Oppenheimer approximation to this problem is quite subtle. Nevertheless, with due care we show that this can be done, as a result we calculate the leading term of the Born-Oppenheimer approximation and indicate how to get the higher order corrections.

math-ph↗

arXiv2016

Born-Oppenheimer Approximation for a Singular System

We discuss a simple singular system in one dimension, two heavy particles interacting with a light particle via an attractive contact interaction. It is natural to apply Born-Oppenheimer approximation to this problem. We present a detailed discussion of this approach, the advantage of this simple model is that one can estimate the error terms self-consistently. Moreover, a Fock space approach to this problem is presented where a systematic expansion can be proposed to get higher order corrections. A slight modification of the same problem in which the light particle is relativistic is discussed in a later section. Here, the second quantized description is more challenging but with some care one can recover the first order term as well as introducing a more systematic approach to higher orders.

math-ph↗

arXiv2015

Infinitely many singular interactions on noncompact manifolds

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan-Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et. al. (2004).

math-ph↗

arXiv2014

Exact Renormalization Group for Point Interactions

Renormalization is one of the deepest ideas in physics, yet its exact implementation in any interesting problem is usually very hard. In the present work, following the approach by Glazek and Maslowski in the flat space, we will study the exact renormalization of the same problem in a nontrivial geometric setting, namely in the two dimensional hyperbolic space. Delta function potential is an asymptotically free quantum mechanical problem which makes it resemble non-abelian gauge theories, yet it can be treated exactly in this nontrivial geometry.

hep-th↗

arXiv2014

Bose-Einstein Condensation on a Manifold with Nonnegative Ricci Curvature

The Bose-Einstein condensation for an ideal Bose gas and for a dilute weakly interacting Bose gas in a manifold with nonnegative Ricci curvature is investigated using the heat kernel and eigenvalue estimates of the Laplace operator. The main focus is on the nonrelativistic gas. However, special relativistic ideal gas is also discussed. The thermodynamic limit of the heat kernel and eigenvalue estimates is taken and the results are used to derive bounds for the depletion coefficient. In the case of a weakly interacting gas Bogoliubov approximation is employed. The ground state is analyzed using heat kernel methods and finite size effects on the ground state energy are proposed. The justification of the c-number substitution on a manifold is given.

math-ph↗