The Liouville Equation for General Ergodic Magnetic Schrödinger Operators
نویسنده
چکیده
Let H(t) ≥ 1 be a time-dependent self-adjoint operator on a Hilbert space H with quadratic form domain Q(H(t)). If Q(H(t)) is independent of t, along with other suitable conditions, we construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation. We apply this extension of Yosida’s Theorem to study the time evolution of a density matrix in a quantum mechanical system, described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, when an electric field is introduced adiabatically. To do this, we introduce and solve a generalized Liouville equation in an appropriate Hilbert space.
منابع مشابه
Linear Response Theory for Magnetic Schrödinger Operators in Disordered Media
We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the non-interacting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of locali...
متن کاملA Liouville-type theorem for Schrödinger operators
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P1, such that a nonzero subsolution of a symmetric nonnegative operator P0 is a ground state. Particularly, if Pj := −∆ + Vj , for j = 0, 1, are two nonnegative Schrödinger operators defined on Ω ⊆ R such that P1 is critical in Ω with a ground state φ, the function ψ 0 is a...
متن کاملOn inverse problem for singular Sturm-Liouville operator with discontinuity conditions
In this study, properties of spectral characteristic are investigated for singular Sturm-Liouville operators in the case where an eigen parameter not only appears in the differential equation but is also linearly contained in the jump conditions. Also Weyl function for considering operator has been defined and the theorems which related to uniqueness of solution of inverse proble...
متن کاملInverse problem for Sturm-Liouville operators with a transmission and parameter dependent boundary conditions
In this manuscript, we consider the inverse problem for non self-adjoint Sturm--Liouville operator $-D^2+q$ with eigenparameter dependent boundary and discontinuity conditions inside a finite closed interval. We prove by defining a new Hilbert space and using spectral data of a kind, the potential function can be uniquely determined by a set of value of eigenfunctions at an interior point and p...
متن کاملLiouville Properties and Critical Value of Fully Nonlinear Elliptic Operators
We prove some Liouville properties for suband supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in OrnsteinUhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully n...
متن کامل