On Spectral Properties of Translationally Invariant Magnetic Schrödinger Operators

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr?dinger operator with such a potential. In particular, we show that the spectrum of is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions exp of the time dependent Schr?dinger equation. It turns out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to an evolution of a one-dimensional free particle but “exits” to +∞ and −∞ in the direction of the potential might be essentially different. Submitted: June 7, 2007. Accepted: August 20, 2007.     

Spectral Properties of Schrödinger Operators on Perturbed Lattices

We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice.     

On the Lipschitz Continuity of Spectral Bands of Harper-Like and Magnetic Schrödinger Operators

We show for a large class of discrete Harper-like and continuous magnetic Schrödinger operators that their band edges are Lipschitz continuous with respect to the intensity of the external constant magnetic field. We generalize a result obtained by Bellissard (Commun Math Phys 160:599–613, 1994), and give examples in favor of a recent conjecture of G. Nenciu.     

On Spectral Problems of Discrete Schrödinger Operators

Applications of Mathematics - A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the...     

Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators

Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.     

Weighted Spectral Gap for Magnetic Schrödinger Operators with a Potential Term

We prove that singular Schrödinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their “nonmagnetic” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L ^p -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrödinger equation with the periodic magnetic field.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

Semiclassical Asymptotics and Gaps in the Spectra of Magnetic Schrödinger Operators

In this paper, we study an L ² version of the semiclassical approximation of magnetic Schrödinger operators with invariant Morse type potentials on covering spaces of compact manifolds. In particular, we are able to establish the existence of an arbitrary large number of gaps in the spectrum of these operators, in the semiclassical limit as the coupling constant goes to zero.     

Pointwise Properties of Eigenfunctions and Heat Kernels of Dirichlet–Schrödinger Operators

Let D be an open domain, a second order elliptic operator with continuous coefficients, and let be a Schrödinger operator associated with H₀, acting on L²(D), with Dirichlet boundary conditions. We provide in this paper both L² and pointwise bounds for the eigenfunctions of H, in term of the Agmon's metric of q and of the quasi-hyperbolic geometry of D. At least when H₀=-, we show that the pointwise bounds obtained for the ground state eigenfunction of H are qualitatively sharp either when q diverges sufficiently fast at the boundary, or in planar regular domains. We also give applications to the intrinsic ultracontractivity of H. Finally, we prove a result concerning the pointwise decay at the boundary of the heat kernel of H in -regular domains.     

Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions

We study spectral properties of Hamiltonians H _X,β,q with δ′-point interactions on a discrete set ${X = \{x_k\}_{k=1}^\infty \subset (0, +\infty)}$ . Using the form approach, we establish analogs of some classical results on operators H _q =  ?d²/dx ² + q with locally integrable potentials ${q \in L^1_{\rm loc}[0, +\infty)}$ . In particular, we establish the analogues of the Glazman–Povzner–Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with δ-interactions, spectral properties of operators H _X,β,q are closely connected with those of ${{\rm H}_{X,q}^N = \oplus_{k}{\rm H}_{q,k}^N}$ , where ${{\rm H}_{q,k}^N}$ is the Neumann realization of ?d²/dx ² + q in L ²(x _k-1,x _k ).     

A Spectral Property of Discrete Schrödinger Operators with Non-Negative Potentials

In the context of an infinite weighted graph of bounded degree, we give a sufficient condition for the discrete Schrödinger operator with a non-negative potential to have a strictly positive bottom of the spectrum. The main result is a discrete analogue of a theorem of Shen in the setting of complete Riemannian manifolds.     

Essential Self-adjointness of Magnetic Schrödinger Operators on Locally Finite Graphs

We give sufficient conditions for essential self-adjointness of magnetic Schrödinger operators on locally finite graphs. Two of the main results of the present paper generalize recent results of Torki-Hamza.     

Absence of Continuous Spectral Types for Certain Non-Stationary Random Schrödinger Operators

We consider continuum random Schrödinger operators of the type H = – + V₀ + V with a deterministic background potential V₀. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of – + V₀. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (random tube) in arbitrary dimension.submitted 07/04/04, accepted 19/08/04     

Sigma Functions and Lie Algebras of Schrödinger Operators

Functional Analysis and Its Applications - In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g > 0$$ , a...     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.     

Spectral deformations of one-dimensional Schrödinger operators

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=?d ²/dx ²+V in $H = - d^2 /dx^2 + V in \mathcal{L}^2 (\mathbb{R})$ . Our technique is connected to Dirichlet data, that is, the spectrum of the operatorH ^D onL ²((?∞,x ₀)) ⊕L ²((x ₀, ∞)) with a Dirichlet boundary condition atx ₀. The transformation moves a single eigenvalue ofH ^D and perhaps flips which side ofx ₀ the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such asV(x)→∞ as |x|→∞, whereV is uniquely determined by the spectrum ofH and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics asE→∞.     

Spectral Multiplicity of Selfadjoint Schrödinger Operators on Star-Graphs with Standard Interface Conditions

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schrödinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of a Theorem of I.S.Kac.     

Commutators of Singular Integral Operators Related to Magnetic Schrdinger Operators

Let A:=-(▽-ia(向量))·(?-ia(向量))+V be a magnetic Schrdinger operator on L~2(R~n),n≥2,where a(向量)=(a_1,···,a_n)∈L~2_(loc)(R~n,R~n) and 0≤V∈L~1_(loc)(R~n).In this paper,we show that for a function b in Lipschitz space Lip_α(R~n) with α∈(0,1),the commutator[b,V~(1/2)A~(-1/2)] is bounded from L_p(R~n) to L_q(R~n),where p,q∈(1,2] and 1/p-1/q =α/n.