Dinger Operator Research Articles

Dispersive Estimates for Schr�dinger Operators in Dimensions One and Three

We consider L^1→L^∞ estimates for the time evolution of Hamiltonians H=−Δ+V in dimensions d=1 and d=3 with bound t^(-∂/2). We require decay of the potentials but no regularity. In d=1 the decay assumption is ∫(1+|x|)|V(x)|dx<∞, whereas in d=3 it is |V(x)|≤C(1+|x|)^(−3−).

Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice $\Z^3$ and interacting via zero-range attractive potentials is considered. For the two-particle energy operator $h(k),$ with $k\in \T^3=(-\pi,\pi]^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of $h(k)$ for $k\neq0$ is proven, provided that $h(0)$ has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr\"{o}dinger operator $H(K), K\in \T^3$ being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number $N(0,z)$ of eigenvalues of H(0) lying below $z<0$ the following limit exists $$ \lim_{z\to 0-} \frac {N(0,z)}{\mid \log\mid z\mid\mid}=\cU_0 $$ with $\cU_0>0$. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum $K$ the finiteness of the number $ N(K,\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the essential spectrum is established and the asymptotics for the number $N(K,0)$ of eigenvalues lying below zero is given.

Spectral Properties of the Periodic Magnetic Schr�dinger Operator in the High-Energy Region. Two-Dimensional Case

The goal is to investigate spectral properties of the operator H=(−i∇ +a(x))2+a0(x) in the two-dimensional situation, a(x), a0(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.

A new characterization of submanifolds with parallel mean curvature vector in $S^{n+p}$

In this work we will consider compact submanifold $M^{n}$ immersed in the Euclidean sphere $S^{n+p}$ with parallel mean curvature vector and we introduce a Schr\{o}dinger operator $L=-\Delta+V$, where $\Delta$ stands for the Laplacian whereas $V$ is some potential on $M^{n}$ which depends on $n,p$ and $h$ that are respectively, the dimension, codimension and mean curvature vector of $M^{n}$. We will present a gap estimate for the first eigenvalue $\mu_{1}$ of $L$, by showing that either $\mu_{1}=0$ or $\mu_{1}\leq-n(1+H^{2})$. As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu \cite{wu} for minimal submanifolds.

An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

We solved the Schrödinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus Tn=Rn/Λ is defined as a quotient of the Euclidean space Rn by an arbitrary n-dimensional lattice Λ. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schrödinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.

Inverse spectral theory for one-dimensional Schr�dinger operators: The A function

We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simon’s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simon’s fundamental equation.

Construction of the Essential Spectrum for a Multidimensional Non-self-adjoint Schr�dinger Operator via the Spectra of Operators with Periodic Potentials, II

Construction of the Essential Spectrum for a Multidimensional Non-self-adjoint Schr�dinger Operator via the Spectra of Operators with Periodic Potentials, II

Construction of the Essential Spectrum for a Multidimensional Non-self-adjoint Schr�dinger Operator via the Spectra of Operators with Periodic Potentials, I

We investigate the topological structure of the essential spectrum\( \sigma_{e}\)(H) of a multidimensional Schrodinger operator H with a complex-valued potential V\(((\mathbf{x})) \) using its description in terms of a family of Schrodinger operators\( \{H\mathbf{^{y}}\}\mathbf{_{y}}\in\mathbb{R}^{m} \) with periodic potentials V\(\mathbf{_{y}}(\mathbf{x})\) which approximate the potential V\( ((\mathbf{x})) \) at infinity in a sense. Under some assumptions on a family of approximating potentials \( \{V_{\mathbf{y}}(\mathbf{X})\}\mathbf{_{y}}\in\mathbb{R}^{m} \), we prove that any compact isolated part of the set \( \sigma_{e}\)(H) consists of a finite number of connected components and for real-valued potentials V\(\mathbf{_{y}}(\mathbf{x})\) the set \( \sigma_{e}\)(H) consists of at most a countable number of segments. For the proof of the last results we develop the theory of awnings. These new topological objects are some kind of fiber bundle, whose fibers are discrete "multiple" sets. We consider several examples of construction of the essential spectrum by using the method developed in this paper.

Eigenvalue asymptotics for the Schr\"odinger operator with a $\delta$-interaction on a punctured surface

Given $n\geq 2$, we put $r=\min\{i\in\mathbb{N}; i>n/2 \}$. Let $\Sigma$ be acompact, $C^{r}$-smooth surface in $\mathbb{R}^{n}$ which contains the origin. Let further $\{S_{\epsilon}\}_{0\le\epsilon<\eta}$ be a family of measurable subsets of $\Sigma$ such that $\sup_{x\in S_{\epsilon}}|x|= {\mathcal O}(\epsilon)$ as $\epsilon\to 0$. We derive an asymptotic expansion for the discrete spectrum of the Schr{\"o}dinger operator $-\Delta -\beta\delta(\cdot-\Sigma \setminus S_{\epsilon})$ in $L^{2}(\mathbb{R}^{n})$, where $\beta$ is a positive constant, as $\epsilon\to 0$. An analogous result is given also for geometrically induced bound states due to a $\delta$ interaction supported by an infinite planar curve.

Scattering and Wave Operators for One-Dimensional Schr�dinger Operators with Slowly Decaying Nonsmooth Potentials

We prove existence of modified wave operators for one-dimensional Schrodinger equations with potential in \(L^p (\mathbb{R}),p < 2.\) If in addition the potential is conditionally integrable, then the usual Moller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.

Asymptotics of the Fourier transform of the spectral measure for Schr$ouml$dinger operators with bounded and unbounded sparse potentials

We study the pointwise behaviour of the Fourier transform of the spectral measure for discrete one-dimensional Schrodinger operators with unbounded sparse potentials, particularly with the potentials of the special type, which give rise to the spectra with Hausdorff dimensionality between 1/2 and 1. Operators with bounded sparse potentials are also considered.

Simple diamagnetic monotonicities for Schr$ouml$dinger operators with inhomogeneous magnetic fields of constant direction

Under certain simplifying conditions we detect monotonicity properties of the ground-state energy and the canonical-equilibrium density matrix of a spinless charged particle in the Euclidean plane subject to a perpendicular, possibly inhomogeneous magnetic field and an additional scalar potential. Firstly, we point out a simple condition warranting that the ground-state energy does not decrease when the magnetic field and/or the potential is increased pointwise. Secondly, we consider the case in which both the magnetic field and the potential are constant along one direction in the plane and give a genuine path-integral argument for corresponding monotonicities of the density-matrix diagonal and the absolute value of certain off-diagonals. Our results complement to some degree results of M. Loss and B. Thaller [Commun. Math. Phys. 186 (1997) 95] and L. Erdos [J. Math. Phys. 38 (1997) 1289].

The spectral bound and principal eigenvalues of Schr�dinger operators on Riemannian manifolds

The spectral bound and principal eigenvalues of Schr�dinger operators on Riemannian manifolds

A short proof of an index theorem

We give a $KK$-theoretical proof of an index theorem for Dirac-Schrödinger operators on a noncompact manifold.

The Absolute Continuity of the Integrated Density¶of States for Magnetic Schrödinger Operators¶with Certain Unbounded Random Potentials

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schr{\"o}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr\"odinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr\"odinger operators with Neumann boundary conditions.

New bounds on the Lieb-Thirring constants

Improved estimates on the constants $L_{\gamma,d}$, for $1/2<\gamma<3/2$, $d\in N$ in the inequalities for the eigenvalue moments of Schr\"{o}dinger operators are established.

Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

We consider a random Schro\"dinger operator in an external magnetic field. The random potential consists of delta functions of random strengths situated on the sites of a regular two-dimensional lattice. We characterize the spectrum in the lowest N Landau bands of this random Hamiltonian when the magnetic field is sufficiently strong, depending on N. We show that the spectrum in these bands is entirely pure point, that the energies coinciding with the Landau levels are infinitely degenerate and that the eigenfunctions corresponding to energies in the remainder of the spectrum are localized with a uniformly bounded localization length. By relating the Hamiltonian to a lattice operator we are able to use the Aizenman-Molchanov method to prove localization.

A Time-Dependent Theory of Quantum Resonances

In this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues. We apply this theory to many-body Schrouml;dinger operators and to the problem of quasiclassical tunneling.

Localization for random perturbations of periodic Schrödinger operators

We prove localization for Anderson-type random perturbations of periodic Schr dinger operators on R near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schr dinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.

On the Spectrum of Odd Order Self-Adjoint Ordinary Differential Operators on the Real Line with Quasi-periodic Coefficients

In this paper we prove that an odd order self-adjoint differential operator on the real line with quasi-periodic coefficients has only absolute continuous simple spectrum for large enough energies. Ordinary differential operators with quasi-periodic coefficients have been studied a lot during the last 20 years. These operators arise naturally from physical considerations. People have mainly been interested in the one dimensional Schro dinger operator with a quasi-periodic potential (see [Be-Te], [Din-Sin], [Eli], [Ji-Si], [Mo], [Mo-Po ], [Su], and [So-Sp]); such hamiltonians come up naturally in solid state physics (see [Pa-Fi]). In this paper, we study ordinary differential operators of higher order with quasi-periodic coefficients. Such operators also come up in physics (for example, in the study of Boussinesq's equation). article no. DE963225