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Fatih Erman

Publications and source records attributed to Fatih Erman.

At least 19 recordsLinked to original sources
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

Bound State Analysis of Rank-One Delta Interactions Supported by Deformed Hyperspheres

We study rank-one delta interactions supported by small normal deformations of an \(n\)-dimensional hypersphere \(S_R^n\subset\mathbb{R}^{n+1}\). The interaction is defined by the normalized surface measure on the support and therefore corresponds to the rotationally invariant sector of the hyperspherical shell problem. For the undeformed hypersphere, we derive the bound-state equation for \(E=-\nu^2\) in terms of the modified Bessel product \[ I_{\frac{n-1}{2}}(\nu R)K_{\frac{n-1}{2}}(\nu R). \] We then consider an inward normal deformation \[ X_\varepsilon(\omega)=(R-\varepsilon h(\omega))\omega, \qquad \omega\in S^n, \qquad 0<\varepsilon\ll1. \] Our main result is that, to first order in \(\varepsilon\), the bound-state energy depends only on the average normal displacement \[ \langle h\rangle = \frac1{|S^n|}\int_{S^n}h(\omega)\,d\Omega_n(\omega). \] Equivalently, the deformed hypersphere is spectrally equivalent, up to \(O(\varepsilon^2)\), to a round hypersphere with effective radius \(R-\varepsilon\langle h\rangle\). In particular, mean-zero deformations do not change the rank-one bound-state energy at first order.

math-ph↗
arXiv2026

Perturbation Theory for Time-dependent Singular Quantum Systems

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. 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.

math-ph↗
arXiv2024

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↗
arXiv2024

The Harmonic Oscillator Potential Perturbed by a Combination of Linear and Non-linear Dirac Delta Interactions with Application to Bose-Einstein Condensation

In this paper, we study the bound state analysis of a one dimensional nonlinear version of the Schr\"{o}dinger equation for the harmonic oscillator potential perturbed by a $\delta$ potential, where the nonlinear term is taken to be proportional to $\delta(x) |\psi(x)|^2 \psi(x)$. The bound state wave functions are explicitly found and the bound state energy of the system is algebraically determined by the solution of an implicit equation. Then, we apply this model to the Bose-Einstein condensation of a Bose gas in a harmonic trap with a dimple potential. We propose that the many-body interactions of the Bose gas can be effectively described by the nonlinear term in the Schr\"{o}dinger equation. Then, we investigate the critical temperature, the condensate fraction, and the density profile of this system numerically.

cond-mat.stat-mech↗
arXiv2023

On Schr\"{o}dinger Operators Modified by $\delta$ Interactions

We study the spectral properties of a Schr\"{o}dinger operator $H_0$ modified by $\delta$ 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 $\delta$ 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 $\delta$ interaction supported on curves, and to the case, where the particle is moving in a compact two-dimensional manifold under the influence of $\delta$ interaction. Finally, the semi-relativistic extension of the last problem has been studied explicitly.

math-ph↗
arXiv2022

Rank One Perturbations Supported by Hybrid Geometries and Their Deformations

We study the hybrid type of rank one perturbations in $\mathbb{R}^2$ and $\mathbb{R}^3$, where the perturbation supported by a circle/sphere is considered together with the delta potential supported by a point outside of the circle/sphere. The construction of the self-adjoint Hamiltonian operator associated with the formal expressions for the rank one perturbation supported by a circle and by a point is explicitly given. The bound state energies and scattering properties for each problem are also studied. Finally, we consider the rank one perturbation supported by a deformed circle/sphere and show that the first order change in the bound state energies under small deformations of the circle/sphere has a simple geometric interpretation. Finally, we consider the delta potentials supported by deformed circle/sphere and show that the first order change in the bound state energies under small deformations of circle/sphere has a simple geometric interpretation.

math-ph↗
arXiv2022

A Direct Method For the Low Energy Scattering Solution of Delta Shell Potentials

A direct method for the bound states and the low energy scattering from a circular and a spherical delta shell potentials is proposed and the results are compared with the one using the standard partial wave analysis developed for potentials with rotational symmetry. The formulation is presented in momentum space and the scattering solutions are obtained by considering the elementary use of distributions. In this approach, the outgoing boundary conditions are imposed explicitly in contrast to the $iε$ prescription often used in quantum mechanics.

math-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↗
arXiv2017

On the Number of Bound States of Point Interactions on Hyperbolic Manifolds

We consider the problem of a quantum particle interacting with $N$ attractive point $δ$-interactions in two and three dimensional Riemannian manifolds and discuss its some spectral properties. The main aim of this paper is to give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds $\mathbb{H}^2$ and $\mathbb{H}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on $\mathbb{R}^2$ and $\mathbb{H}^2$.

math-ph↗
arXiv2017

One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials

In this paper, we consider the one-dimensional semirelativistic Schrödinger equation for a particle interacting with $N$ Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an $N \times N$ matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most $N$ bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a self-adjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for $N$ centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two-center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.

math-ph↗
arXiv2016

Recursion formula for the Green's function of a Hamiltonian for several types of Dirac delta-function potentials in curved spaces

In this short article, we non-perturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We also extend this formula to the case for the Dirac delta potentials supported by regular curves embedded in two dimensional manifolds and for the Dirac delta potentials supported by two dimensional compact manifolds embedded in three dimensional manifolds. Finally, this formulation allows us to find the recursive formula of the Green's function for the point Dirac delta potentials in two and three dimensional Riemannian manifolds, where the renormalization of coupling constant is required.

math-ph↗
arXiv2014

Nondegeneracy of the Ground State for Nonrelativistic Lee Model

In the present work, we first briefly sketch construction of the nonrelativistic Lee model on Riemannian manifolds, introduced in our previous works. In this approach, the renormalized resolvent of the system is expressed in terms of a well-defined operator, called the principal operator, so as to obtain a finite formulation. Then, we show that the ground state of the nonrelativistic Lee model on a compact Riemannian manifolds is nondegenerate using the explicit expression of the principal operator that we obtained. This is achieved by combining heat kernel methods with positivity improving semi-group approach and then applying these tools directly to the principal operator, rather than the Hamiltonian, without using cut-offs.

math-ph↗
arXiv2012

A Many-body Problem with Point Interactions on Two Dimensional Manifolds

A non-perturbative renormalization of a many-body problem, where non-relativistic bosons living on a two dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan-Symanzik equation) for the principal operator of the model is derived and the $β$ function is exactly calculated for the general case, which includes all particle numbers.

math-ph↗
arXiv2011

Existence of Hamiltonians for Some Singular Interactions on Manifolds

The existence of the Hamiltonians of the renormalized point interactions in two and three dimensional Riemannian manifolds and that of a relativistic extension of this model in two dimensions are proven. Although it is much more difficult, the proof of existence of the Hamiltonian for the renormalized resolvent for the non-relativistic Lee model can still be given. To accomplish these results directly from the resolvent formula, we employ some basic tools from the semigroup theory.

math-ph↗
arXiv2011

Non-relativistic Lee Model on two Dimensional Riemannian Manifolds

This work is a continuation of our previous work (JMP, Vol. 48, 12, pp. 122103-1-122103-20, 2007), where we constructed the non-relativistic Lee model in three dimensional Riemannian manifolds. Here we renormalize the two dimensional version by using the same methods and the results are shortly given since the calculations are basically the same as in the three dimensional model. We also show that the ground state energy is bounded from below due to the upper bound of the heat kernel for compact and Cartan-Hadamard manifolds. In contrast to the construction of the model and the proof of the lower bound of the ground state energy, the mean field approximation to the two dimensional model is not similar to the one in three dimensions and it requires a deeper analysis, which is the main result of this paper.

math-ph↗
arXiv2010

Point Interaction in two and three dimensional Riemannian Manifolds

We present a non-perturbative renormalization of the bound state problem of n bosons interacting with finitely many Dirac delta interactions on two and three dimensional Riemannian manifolds using the heat kernel. We formulate the problem in terms of a new operator called the principal or characteristic operator. In order to investigate the problem in more detail, we then restrict the problem to one particle sector. The lower bound of the ground state energy is found for general class of manifolds, e.g., for compact and Cartan-Hadamard manifolds. The estimate of the bound state energies in the tunneling regime is calculated by perturbation theory. Non-degeneracy and uniqueness of the ground state is proven by Perron-Frobenius theorem. Moreover, the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics. Renormalization procedure does not lead to any radical change in these cases. Finally, renormalization group equations are derived and the beta-function is exactly calculated. This work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions, developed by S. G. Rajeev.

math-ph↗