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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)

Renormalization is one of the deepest ideas in physics, yet its exact implementation in any interesting problem is usually very hard. In the present work, following the approach by Glazek and Maslowski in the flat space, we will study the exact renormalization of the same problem in a nontrivial geometric setting, namely in the two dimensional hyperbolic space. Delta function potential is an asymptotically free quantum mechanical problem which makes it resemble nonabelian gauge theories, yet it can be treated exactly in this nontrivial geometry.

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

The mixed-resolution elastic network model was introduced previously for computing the motions of a structure, which is described at different levels of detail in different parts, for example, with atomistic and residue-level regions. This method has proved to be an efficient tool to explore the collective dynamics of proteins with some atomistic details, which would be difficult to obtain with either conventional full-atom approaches or fully coarse-grained models. Understanding function often requires atomic detail, but not necessarily for the entire structure. In this study, the calculation of the interaction forces between different resolution regions for the hierarchical levels of coarse-graining is further elaborated on in the new approach by considering explicitly the atomic contacts in the crystal structure. The collective dynamics of the enzyme triosephosphate isomerase and its active site together with loop 6 motions are considered in detail. The supramolecular assemblage ribosome and local atomic motions in its “interesting” functional part—the decoding center—are investigated for the low frequency range of the spectrum with high computational efficiency. This new atom-based mixed coarse-graining approach can be effectively used to generate realistic high-resolution conformations of extremely large protein-DNA or RNA complexes by performing energy minimization on structures deformed along the normal modes of the elastic network model. The new model permits focusing on specific functional parts that move in coordination and response to the remainder of the entire structure.