Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2409.05372 · Frontiers in Physics

Completeness relation in renormalized quantum systems

Fatih Erman, O. Teoman Turgut

Published Nov 13, 2025• 1 citation
arXivHTMLPDF

Abstract

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 an 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 the sudden perturbation of the support of the delta potential.

DOI
10.3389/fphy.2025.1695365

Related papers

Renormalized contact potential in two dimensions

R. J. Henderson, S. G. Rajeev

We obtain for the attractive Dirac δ-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N−2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N−2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The three-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given.

arXiv:hep-th/9710061•Journal of Mathematical Physics•Feb 1, 1998•11 citations

Perturbation Theory for Time-Dependent Point Interactions with Discrete Spectrum

Kaya Güven Akbaş, Fatih Erman, O. Teoman Turgut

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-phCross-listedarXiv:2608.08108•Aug 11, 2026•0 citations

RENORMALIZED CONTACT POTENTIAL IN TWO DIMENSIONS

Robert Henderson, S. G. Rajeev

We obtain for the attractive Dirac δ-function potential in twodimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N −2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N − 2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The 3-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given. I.

•Jan 1, 1997•8 citations

Renormalized path integral for the two-dimensional δ-function interaction

Horacio E. Camblong, CARLOS R. ORDÓÑEZ

A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.

arXiv:hep-th/0110176•Physical Review A•May 17, 2002•30 citations

Three Identical One-Dimensional Quantum Particles with Point Interaction as a Solvable Model: I. Discrete Spectrum

M. A. Lyalinov

The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with $$ \delta $$ -functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions. DOI 10.1134/S1061920825600850

Russian Journal of Mathematical Physics•Sep 1, 2025•2 citations

Exactly solvable models of sphere interactions in quantum mechanics

Jean-Pierre Antoine, Fritz Gesztesy, J Shabani

The authors discuss exactly solvable Schrodinger Hamiltonians corresponding to a surface delta interaction supported by a sphere and various generalisations thereof. First they treat the pure delta sphere model; self-adjointness of the Hamiltonian, spectral properties, stationary scattering theory, approximation by scaled short-range Hamiltonians. Next they extend the model by adding a point interaction at the centre of the sphere or, alternatively, a Coulomb interaction. Finally the whole analysis is extended to the case of a delta ' sphere interaction, first taken alone, then superimposed on a point interaction or a Coulomb potential.

Journal of Physics A Mathematical and General•Aug 21, 1987•114 citations
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 an 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 the sudden perturbation of the support of the delta potential.
We obtain for the attractive Dirac δ-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N−2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N−2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The three-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given.
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.
We obtain for the attractive Dirac δ-function potential in twodimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N −2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N − 2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The 3-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given. I.
A path-integral approach for \ensuremath{\delta}-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale-invariant problem in quantum mechanics. Our treatment is based on an infinite summation of perturbation theory that captures the nonperturbative nature of the \ensuremath{\delta}-function bound state. The well-known singular character of the two-dimensional \ensuremath{\delta}-function potential is dealt with by considering the renormalized path integral resulting from a variety of schemes: dimensional, momentum-cutoff, and real-space regularization. Moreover, compatibility of the bound-state and scattering sectors is shown.
The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with δ \delta δ -functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions. DOI 10.1134/S1061920825600850
The authors discuss exactly solvable Schrodinger Hamiltonians corresponding to a surface delta interaction supported by a sphere and various generalisations thereof. First they treat the pure delta sphere model; self-adjointness of the Hamiltonian, spectral properties, stationary scattering theory, approximation by scaled short-range Hamiltonians. Next they extend the model by adding a point interaction at the centre of the sphere or, alternatively, a Coulomb interaction. Finally the whole analysis is extended to the case of a delta ' sphere interaction, first taken alone, then superimposed on a point interaction or a Coulomb potential.