We analyze the 1 + 1 dimensional Nambu-Jona-Lasinio model nonperturbatively. In addition to its simple ground state saddle points, the effective action of this model has a rich collection of non-trivial saddle points in which the composite fields σ(x) = 〈ψ̄ψ〉 and π(x) = 〈ψ̄iγ5ψ〉 form static space dependent configurations because of non-trivial dynamics. These configurations may be viewed as one d...

We consider scattering by general compactly supported semi-classical perturbations of the Euclidean Laplace-Beltrami operator. We show that if the suitably cut-off resolvent of the Hamiltonian quantizes a Lagrangian relation on the product cotangent bundle, the scattering amplitude quantizes the natural scattering relation. In the case when the resolvent is tempered, which is true under some no...

We construct the symmetric functional model for an arbitrary closed operator with a non-empty resolvent set acting on a separable Hilbert space. The main techniques of the study are based on the explicit form of the Sz.-Nagy-Foiaş model for a closed dissipative operator, the Potapov-Ginzburg transform of characteristic functions, and certain resolvent identities. All considerations are carried ...

Transmission eigenvalues are points in the spectrum of the interior transmission operator, a coupled 2x2 system of elliptic partial differential equations, where one unknown function must satisfy two boundary conditions and the other must satisfy none. We show that the interior transmission eigenvalues are discrete and depend continuously on the contrast by proving that the interior transmissio...

In this paper, we introduce a new class of variational inclusions involving three operator. Using the resolvent operator technique, we establish the equivalence between the general variational inclusions and the resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general variational inclusions. We also consider the cr...

We discuss the problem of deriving estimates for the resolvent of the linear operator associated with three dimensional perturbations of plane Couette flow, and determining its dependence on the Reynolds number R. Depending on the values of the parameters involved, we derive estimates analytically. For the remaining values of the parameters, we prove that deriving estimates for the resolvent ca...