We study the semi-classical behavior of the spectral function of the Schrödinger operator with short range potential. We prove that the spectral function is a semiclassical Fourier integral operator quantizing the forward and backward Hamiltonian flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spe...

This paper studies previously developed nonlinear Hilbert adjoint operator theory from a variational point of view and provides a formal justification for the use of Hamiltonian extensions via Gâteaux differentials. The primary motivation is its use in characterizing singular values of nonlinear operators, and in particular, the Hankel operator and its relationship to the state space notion of ...

The time operator for a quantum singular oscillator of the Calogero-Sutherland type is constructed in terms of the generators of the SU(1,1) group. In the space spanned by the eigenstates of the Hamiltonian, the time operator is not self-adjoint. We show, that the time-energy uncertainty relation can be given the meaning within the Barut-Girardello coherent states defined for the singular oscil...

It is shown that an operator can be defined in the abstract space of a random matrix ensemble whose matrix elements statistical distribution simulates the behavior of the distribution found in real physical systems. It is found that the key quantity that determine these distributions is commutator of the operator with the Hamiltonian. Application to symmetry breaking in quantum many-body system...

In this article we prove that the wave operators describing the direct scattering of the defocusing matrix Zakharov-Shabat system with potentials having distinct nonzero values with the same modulus at ±∞ exist, are asymptotically complete, and lead to a unitary scattering operator. We also prove that the free Hamiltonian operator is absolutely continuous.

Assuming the Lorentz and CPT invariances we show that neutron-antineutron oscillation implies breaking of CP along with baryon number violation – i.e. two of Sakharov conditions for baryogenesis. The oscillation is produced by the unique operator in the effective Hamiltonian. This operator mixing neutron and antineutron preserves charge conjugation C and breaks P and T. External magnetic field ...

We examine the multiple Hamiltonian structure and construct a symplectic realization of the Volterra model. We rediscover the hierarchy of invariants, Poisson brackets and master symmetries via the use of a recursion operator. The rational Volterra bracket is obtained using a negative recursion operator.

The |∆I| = 1/2 rule in non-leptonic decays of hyperons can be naturally understood by postulating a priori mixed physical hadrons, along with the isospin invariance of the responsible transition operator. It is shown that this operator can be identified with the strong interaction Yukawa hamiltonian. The experimental amplitudes are well reproduced.

${\cal C}$-operators were introduced as involution operators in non-Hermitian theories that commute with the time-independent Hamiltonians and parity/time-reversal operator. Here we propose a definition for time-dependent C}(t)$-operators demonstrate particular signature they may be expanded terms of biorthonormal left right eigenvectors Lewis-Riesenfeld invariants. The vanishing commutation re...

We consider the quantum graph Hamiltonian on square lattice in Euclidean space, and we show that spectrum of converges to corresponding Schr\"odinger operator space continuum limit, eigenfunctions eigenprojections also converge some sense. employ discrete as intermediate operator, use a recent result by second third author limit operator.