The spectral problem (A+V (z))ψ = zψ is considered with A, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation V (z) is assumed to depend on the energy z as resolvent of another self-adjoint operator A ′ : V (z) = −B(A ′ −z) −1 B *. It is supposed that operator B has a finite Hilbert-Schmidt norm and spectra of operators A and A ′ are separated. The conditions are form...

This is the continuation of a previous article in which the Bjorken and Voloshin sum rules were interpreted as statements of conservation of probability and energy. Here the formalism is extended to higher moments of the Hamiltonian operator. From the conservation of the second moment of the Hamiltonian operator one can derive a sum rule which, in the small velocity limit, reduces to the Bigi-G...

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of canonical Hamiltonian systems. Traditional intrusive projection-based model reduction approaches utilize symplectic Galerkin projection construct ROMs by projecting Hamilton’s equations the full onto subspace. requires complete knowledge about operators and access manipulate computer code. In con...

We construct the Hamiltonian operator of the string field theory for c = 0 string theory. It describes how strings evolve in the coordinate frame, which is defined by using the geodesic distance on the worldsheet. The Hamiltonian consists of three-string interaction terms and a tadpole term. We show that one can derive the loop amplitudes of c = 0 string theory from this Hamiltonian.

We prove the integrability of the short pulse equation derived recently by Schäfer and Wayne from a hamiltonian point of view. We give its bi-hamiltonian structure and show how the recursion operator defined by the hamiltonian operators is connected with the one obtained by Sakovich and Sakovich. An alternative zero-curvature formulation is also given. PACS: 02.30.Ik; 02.30.Jr; 05.45.-a

We show that under minor technical assumptions any weakly nonlocal Hamiltonian structures compatible with a given nondegenerate weakly nonlocal symplectic operator J can be written as the Lie derivatives of J along a suitably chosen nonlocal vector field. Moreover, we present a new description for local Hamiltonian structures of arbitrary order compatible with a given nondegenerate local Hamilt...

In this paper, I consider one-dimensional periodic Schrödinger operators perturbed by a slowly decaying potential. In the adiabatic limit, I give an asymptotic expansion of the eigenvalues in the gaps of the periodic operator. When one slides the perturbation along the periodic potential, these eigenvalues oscillate. I compute the exponentially small amplitude of the oscillations.

Asymptotics at large time of the Green function to the wave equation with periodic coeecients are found. A particular attention is given to its asymptotics near the wave front. It is shown that the spectral band (with number n = 0; 1; 2; ::) of the corresponding Hill operator is associated with a wave having the front velocity c n < 1. Estimates for c n in the terms of the gap lengths, the eeec...

An analysis of extension of Hamiltonian operators from lower order to higher order of matrix paves a way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a consequence of bi-Hamiltonian structure, they all possess infinitely many symmetrie...

We study the distribution of eigenvalues of the Schrödinger operator with a complex valued potential V . We prove that if |V | decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.