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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., Comment: 24 pages, 4 figures

In a recent paper, a variational principle for Liouville dynamics (vector fields preserving a volume form) was introduced. In the present paper, we discuss the relation between symmetries and conservation laws in Liouville dynamics. We recall a result by Crampin relating symmetries and conserved quantities in Liouville dynamics, and discuss how symmetry reduction is implemented in the frame of the variational principle for Liouville dynamics.

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., Comment: 16 pages, typos are corrected, abstract and some sentences in the text are improved. Appears in Journal of Mathematical Physics, volume 55, issue 8 (2014)

We study the relativistic Lee model on static Riemannian manifolds. The model is constructed nonperturbatively through its resolvent, which is based on the so-called principal operator and the heat kernel techniques. It is shown that making the principal operator well-defined dictates how to renormalize the parameters of the model. The renormalization of the parameters are the same in the light front coordinates as in the instant form. Moreover, the renormalization of the model on Riemannian manifolds agrees with the flat case. The asymptotic behavior of the renormalized principal operator in the large number of bosons limit implies that the ground state energy is positive. In 2+1 dimensions, the model requires only a mass renormalization. We obtain rigorous bounds on the ground state energy for the n-particle sector of 2+1 dimensional model., 23 pages, added a new section, corrected typos and slightly different title