Perturbation theory of eigenvalue problems

Perturbation of eigenvalues and scattering resonances with applications in imaging

This paper presents an investigation on the global hypoellipticity problem for a class of systems of pseudo‐differential operators on the torus. The approach consists in establishing conditions on the matrix symbol of the system such that it can be transformed into a suitable triangular form involving a nilpotent upper triangular matrix. Hence, the global hypoellipticity is studied by analyzing the behavior of the eigenvalues and their averages.

Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

The application of Laplace transformation to a class of non-stationary problems of mathematical physics leads to analysis of non-selfadjoint regular elliptic operators. It is shown that these operators have in the right-hand side halfplane of the transform parameter discrete regular spectrum and eigenelements form a complete set biorthonormal with eigenelements of the adjoint operator which are complex conjugate. Finally finite element solution is proposed.

Spectral Analysis of Non-Selfadjoint Operators of Mathematical Physics

The RSPT (Chapter III) allows one to get an approximation to the eigenvalues (En) of a given Hamiltonian operator through a series in powers of a real parameter λ. However, the usefulness of the power series is conditioned by a fundamental question: its convergence.

Divergence of the Perturbation Series

We define the quantum $p$-divergences and introduce Beckner's inequalities for primitive quantum Markov semigroups on a finite-dimensional matrix algebra satisfying the detailed balance condition. Such inequalities quantify the convergence rate of the quantum dynamics in the noncommutative $L_p$-norm. We obtain a number of implications between Beckner's inequalities and other quantum functional inequalities, as well as the hypercontractivity. In particular, we show that the quantum Beckner's inequalities interpolate between the Sobolev-type inequalities and the Poincar\'{e} inequality in a sharp way. We provide a uniform lower bound for the Beckner constant $\alpha_p$ in terms of the spectral gap and establish the stability of $\alpha_p$ with respect to the invariant state. As applications, we compute the Beckner constant for the depolarizing semigroup and discuss the mixing time. For symmetric quantum Markov semigroups, we derive the moment estimate, which further implies a concentration inequality. We introduce a new class of quantum transport distances $W_{2,p}$ interpolating the quantum 2-Wasserstein distance by Carlen and Maas [J. Funct. Anal. 273(5), 1810-1869 (2017)] and a noncommutative $\dot{H}^{-1}$ Sobolev distance. We show that the quantum Markov semigroup with $\sigma$-GNS detailed balance is the gradient flow of a quantum $p$-divergence with respect to the metric $W_{2,p}$. We prove that the set of quantum states equipped with $W_{2,p}$ is a complete geodesic space. We then consider the associated entropic Ricci curvature lower bound via the geodesic convexity of $p$-divergence, and obtain an HWI-type interpolation inequality. This enables us to prove that the positive Ricci curvature implies the quantum Beckner's inequality, from which a transport cost and Poincar\'{e} inequalities can follow.

Perturbation theory of eigenvalue problems

Citations

Perturbation of eigenvalues and scattering resonances with applications in imaging

Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

Spectral Analysis of Non-Selfadjoint Operators of Mathematical Physics

Divergence of the Perturbation Series

Interpolation between modified logarithmic Sobolev and Poincare inequalities for quantum Markovian dynamics

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