Nonrelativistic Lee model in three dimensional Riemannian manifolds

Erman, Fatih; Turgut, O. Teoman

doi:10.1063/1.2813026

Cited by 7 publications

(23 citation statements)

References 17 publications

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“…We have also proved in [4,5,11] that the Hamiltonian is bounded from below and this lower bound is given by…”

Section: Ground Statementioning

confidence: 93%

“…In this work, we will generalize the arguments developed for the point interactions to prove the nondegeneracy of the ground state of the Lee model defined on Riemannian manifolds, the construction of which were already established in our previous studies [4,5] by extending the ideas introduced in [30]. Although the basic idea in proving the nondegeneracy of the ground state in this model is similar to the one which we developed for point interactions, the proof requires the use of the positivity arguments.…”

Section: Introductionmentioning

confidence: 94%

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Nondegeneracy of the ground state for nonrelativistic Lee model

Erman

Malkoç

Turgut

2014

Journal of Mathematical Physics

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In 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.

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“…We have also proved in [4,5,11] that the Hamiltonian is bounded from below and this lower bound is given by…”

Section: Ground Statementioning

confidence: 93%

Section: Introductionmentioning

confidence: 94%

Nondegeneracy of the ground state for nonrelativistic Lee model

Erman

Malkoç

Turgut

2014

Journal of Mathematical Physics

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“…Now, in order to see and separate out the divergent part from (20), we will normal order the operators in (20) by using the commutation relations of the field operators. For simplicity, we explicitly perform our calculations for compact manifolds here, but our result is also valid, in principle, for non-compact manifolds by using a similar method to that we have used for the non-relativistic Lee model [37,38].…”

Section: Construction Of the Modelmentioning

confidence: 99%

“…which are the generalized versions of equations we first used in [37]. Therefore, from equation (89), we read the state vector | 0 of our many-body system in terms of the eigenstate |φ 0 of the principal operator…”

Section: φ(E ) φ(E )| = |ω(E ); (E ) ω(E ); (E )|mentioning

confidence: 99%

“…Let us give a better justification of this choice: we will again assume that the orthofermion operators act on some smooth functions; since the set of smooth functions is dense in the Hilbert space norm, this is allowed.We will write one of the heat kernels as a distributional solution in (37), and use the fact that −∇ 2 g is a self-adjoint operator,…”

Section: Construction Of the Modelmentioning

confidence: 99%

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A many-body problem with point interactions on two-dimensional manifolds

Erman¹,

Turgut²

2013

J. Phys. A: Math. Theor.

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A non-perturbative renormalization of a many-body problem, where nonrelativistic 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.

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Renormalized interaction of relativistic bosons with delta function potentials

Doǧan

Turgut

2010

Journal of Mathematical Physics

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We study the interaction of mutually noninteracting Klein–Gordon particles with localized sources on stochastically complete Riemannian surfaces. This asymptotically free theory requires regularization and coupling constant renormalization. Renormalization is performed nonperturbatively using the orthofermion algebra technique and the principal operator Φ is found. The principal operator is then used to obtain the bound state spectrum, in terms of binding energies to single Dirac-delta function centers. The heat kernel method allows us to generalize this procedure to compact and Cartan–Hadamard type Riemannian manifolds. We make use of upper and lower bounds on the heat kernel to constrain the ground state energy from below, thus confirming that our neglect of pair creation is justified for certain ranges of parameters in the problem.

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Nonrelativistic Lee model in three dimensional Riemannian manifolds

Cited by 7 publications

References 17 publications

Nondegeneracy of the ground state for nonrelativistic Lee model

Nondegeneracy of the ground state for nonrelativistic Lee model

A many-body problem with point interactions on two-dimensional manifolds

Renormalized interaction of relativistic bosons with delta function potentials

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