922 research outputs found
On the number of bound states of point interactions on hyperbolic manifolds
No full textWe study the bound state problem for N attractive point Dirac δ-interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have N bound states and give an explicit criterion for it in hyperbolic manifolds ℍ2 and ℍ3. Furthermore, we study the same spectral problem for a relativistic extension of the model on ℝ2 and ℍ2
Renormalization of Dirac delta potentials through minimal extension of Heisenberg algebra
No full textWe renormalize the model of multiple Dirac delta potentials in two and three dimensions by regularizing it through the minimal extension of Heisenberg algebra. We show that the results are consistent with the other regularization schemes given in the literature
Recursion formula for the Green's function of a Hamiltonian for several types of Dirac delta-function potentials in curved spaces
No full textIn this short article, we nonperturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We extend this formula to the one 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
A perturbative approach to the tunneling phenomena
No full textThe 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 path-integral, WKB, and instanton calculations. All these methods are non-perturbative and there is a common belief that it is dif fi cult to fi nd the splitting in the energy due to the barrier penetration from a perturbative analysis. However, we will illustrate by explicit examples including singular potentials (e.g., Dirac delta potentials supported by points and curves and their relativistic extensions) it is possible to fi nd the splitting in the bound state energies by developing some kind of perturbation method
A many-body problem with point interactions on two-dimensional manifolds
No full textA 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 beta function is exactly calculated for the general case, which includes all particle numbers
On Scattering from the One Dimensional Multiple Dirac Delta Potentials
No full textFísica Teórica. Atómica y Óptic
Existence of Hamiltonians for some singular interactions on manifolds
No full textThe 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
Nondegeneracy of the ground state for nonrelativistic Lee model
No full textIn the present work, we first briefly sketch the 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 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
A singular one-dimensional bound state problem and its degeneracies
No full textProducción CientíficaWe give a brief exposition of the formulation of the bound state problem for the one-dimensional system of N attractive Dirac delta potentials, as an N×N matrix eigenvalue problem (ΦA=ωA). The main aim of this paper is to illustrate that the non-degeneracy theorem in one dimension breaks down for the equidistantly distributed Dirac delta potential, where the matrix Φ becomes a special form of the circulant matrix. We then give elementary proof that the ground state is always non-degenerate and the associated wave function may be chosen to be positive by using the Perron-Frobenius theorem. We also prove that removing a single center from the system of N delta centers shifts all the bound state energy levels upward as a simple consequence of the Cauchy interlacing theorem.Física Teórica. Atómica y Óptic
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