Schrodinger Operator Research Articles

Weighted Inequalities for Schrödinger Type Singular Integrals

Related to the Schrodinger operator $$L=-\Delta +V$$ , the behaviour on $$L^p$$ of several first and second order Riesz transforms was studied by Shen (Ann Inst Fourier (Grenoble) 45(2):513–546, 1995). Under his assumptions on V, a critical radius function $$\rho :X\rightarrow {\mathbb {R}}^+$$ can be associated, with the property that its variation is controlled by powers. Given such a function, we introduce a class of singular integral operators whose kernels have some extra decay related to $$\rho $$ . We analyse their behaviour on weighted $$L^p$$ and BMO-type spaces. Here, the weights as well as the regularity spaces depend only on the critical radius function. When our results are set back into the Schrodinger context, we obtain weighted inequalities for all the Riesz transforms initially appearing in Shen (1995). Concerning the action of Schrodinger singular integrals on regularity spaces, we extend some previous work of Ma et al.

Fundamental matrices and Green matrices for non-homogeneous elliptic systems

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in Rn and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi– Nash–Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schr¨odinger (−∆ + V ) and generalized Schrodinger (− div A∇ + V ) operators with real and complex coefficients, on arbitrary domains.

Spectrum of Schrödinger Hamiltonian operator with singular inverted complex and Kratzer’s molecular potentials in fractional dimensions

Singular potentials play a key role in the study of quantum properties of molecular interactions and in different branches of physics and quantum chemistry. They assist us to understand the structure of condensed matter and several biological dynamical systems as well as a number of chemical processes. Complex-potential models arise also in nuclear, atomic molecular physics and other fields, and are of special interest. Most of the studies done in the literature are based on the analysis of quantum systems with integer dimensions. However, the concept of fractional or non-integer dimensions has received recently much interest, since a number of quantum physics phenomena are accurately modelled in fractional dimensional spaces. In this paper, we determine the spectrum of the Schrodinger operator in fractional dimensions with an inverted complex singular potential and we solve the corresponding time-dependent wave equation for the case of a complex singular potential and a Kratzer’s molecular potential, which has wide applications in solid-state physics and molecular physics. Several properties are analyzed and discussed accordingly.

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrodinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrodinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrodinger operators with periodic potentials. For quasi-convex domains in Rn, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.

Exact and Approximate Solutions of Spectral Problems for the Schrödinger Operator on (−∞, ∞) with Polynomial Potential

New exact representations for the solutions of numerous one-dimensional spectral problems for the Schrodinger operator with polynomial potential are obtained by using a technique based on the functional-discrete (FD) method. In the cases where the ordinary FD-method is divergent, we propose to use its modification, which proves to be quite efficient. The obtained theoretical results are illustrated by numerical examples.

On expansion of functions from the S.M. Nikolskii class in a Fourier series by eigenfunctions of the Schrodinger operator

On expansion of functions from the S.M. Nikolskii class in a Fourier series by eigenfunctions of the Schrodinger operator

Efficient Spectral Methods for Some Singular Eigenvalue Problems

We propose and analyze some efficient spectral/spectral element methods to solve singular eigenvalue problems related to the Schrodinger operator with an inverse-power potential. For the Schrodinger eigenvalue problem $$-\Delta u +V(x)u=\lambda u$$ with a regular potential $$V(x)=c_1|x|^{-1}$$ , we first design an efficient spectral method on a ball of any dimension by adopting the Sobloev-orthogonal basis functions with respect to the Laplacian operator to overwhelm the homogeneous inverse potential and to eliminate the singularity of the eigenfunctions. Then we extend this spectral method to arbitrary polygonal domains by the mortar element method with each corner covered by a circular sector and origin covered by a circular disc. Furthermore, for the Schrodinger eigenvalue problem with a singular potential $$V(x)=c_3|x|^{-3}$$ , we devise a novel spectral method by modifying the former Sobloev-orthogonal bases to fit the stronger singularity. As in the case of $$|x|^{-1}$$ potential, this approach can be extended to arbitrary polygonal domains by the mortar element method as well. Finally, for the singular elliptic eigenvalue problem $$-\frac{\partial ^2}{\partial x^2}u-\frac{1}{x^2}\frac{\partial ^2}{\partial y^2}u =\lambda u$$ on rectangles, we propose a novel spectral method by using tensorial bases composed of the $$L^2$$ - and $$H^1$$ -simultaneously orthogonal functions in the y-direction and the Sobolev-orthogonal functions with respect to the Schrodinger operator with an inverse-square potential in the x-direction. Numerical experiments indicate that all our methods possess exponential orders of convergence, and are superior to the existing polynomial based spectral/spectral element methods and hp-adaptive methods.

Criticality theory for Schrödinger operators with singular potential

This paper extends a weighted spectral gap classification for Schrodinger operators −Δ+V to a large class of locally integrable potentials V called balanced potentials here. Harnack's inequality will not hold in general for such potentials. In addition to the standard potentials in the space Llocp for some p>N2, this class contains potentials for which V+ is locally bounded (with no restriction on V−) or more generally, those for which the positive and negative singularities are separated as well as those for which the interaction of the positive and negative singularities is not “too strong”.We also establish for such potentials, under the additional assumption that V+∈Llocp for some p>N2, versions of the Agmon–Allegretto–Piepenbrink Principle characterising the subcritical/critical operators in terms of the cone of positive distributional super-solutions.

The Lowest Eigenvalue of Schrödinger Operators on Compact Manifolds

The lowest eigenvalue of the Schrodinger operator $-{\Delta }+\mathcal {V}$ on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.

О спектре двумерного оператора Шрёдингера с однородным магнитным полем и периодическим электрическим потенциалом

We consider the two-dimensional Schrodinger operator $\widehat H_B+V$ with a uniform magnetic field $B$ and a periodic electric potential $V$. The absence of eigenvalues (of infinite multiplicity) in the spectrum of the operator $\widehat H_B+V$ is proved if the electric potential $V$ is a nonconstant trigonometric polynomial and the condition $(2\pi )^{-1}\, Bv(K)=Q^{-1}$ for the magnetic flux is fulfilled where $Q\in \mathbb{N}$ and the $v(K)$ is the area of the elementary cell $K$ of the period lattice $\Lambda \subset \mathbb{R}^2$ of the potential $V$. In this case the singular component of the spectrum is absent so the spectrum is absolutely continuous. In this paper, we use the magnetic Bloch theory. Instead of the lattice $\Lambda $ we choose the lattice $\Lambda _{\, Q}=\{ N_1QE^1+N_2E^2:N_j\in \mathbb{Z} , j=1,2\} $ where $E^1$ and $E^2$ are basis vectors of the lattice $\Lambda $. The operator $\widehat H_B+V$ is unitarily equivalent to the direct integral of the operators $\widehat H_B(k)+V$ with $k\in 2\pi K_{\, Q}^*$ acting on the space of magnetic Bloch functions where $K_{\, Q}^*$ is an elementary cell of the reciprocal lattice $\Lambda _{\, Q}^*\subset \mathbb{R}^2$. The proof of the absence of eigenvalues in the spectrum of the operator $\widehat H_B+V$ is based on the following assertion: if $\lambda $ is an eigenvalue of the operator $\widehat H_B+V$, then the $\lambda $ is an eigenvalue of the operators $\widehat H_B(k+i\varkappa )+V$ for all $k,\, \varkappa \in \mathbb{R}^2$ and, moreover, (under the assumed conditions on the $V$ and the $B$) there is a vector $k_0\in \mathbb{C}^2\, \backslash \, \{0\}$ such that the eigenfunctions of the operators $\widehat H_B(k+\zeta k_0)+V$ with $\zeta \in \mathbb{C}$ are trigonometric polynomials $\sum \zeta ^j\Phi _j$ in $\zeta $.

A One-Dimensional Schrödinger Operator with Square-Integrable Potential

We study the spectral properties of a one-dimensional Schrodinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.

Monotonicity of Eigenvalues and Functionals Along the Ricci–Bourguignon Flow

In this paper we first prove the monotonicity of the lowest eigenvalue of the Schrodinger operator $$\begin{aligned} \frac{(1-(n-1)\rho )^2}{1-2(n-1)\rho }R-4\varDelta \end{aligned}$$ along the Ricci–Bourguignon flow $$\begin{aligned} \frac{\partial \mathrm {g}}{\partial t}=-2(\mathrm {Ric}-\rho R\mathrm {g}) \end{aligned}$$ based on an evolving formula of the $$\mathcal {F}_{\rho }$$ -functional, and rule out nontrivial steady breathers. Then we prove the monotonicity of the lowest eigenvalue of the Schrodinger operator $$BR-4\varDelta $$ along the Ricci–Bourguignon flow for any constant B satisfying $$\begin{aligned} B\ge \frac{4(1-(n-1)\rho )^2-n\rho }{4(1-2(n-1)\rho )}>0 \end{aligned}$$ for the case that $$\rho \le 0.$$ We also study the evolving formula of the $$\mathcal {W}_{\rho }$$ -functional and get the monotonicity of the infimum of the $$\mathcal {W}_{\rho }$$ -functional, based on which we can prove that a shrinking breather should be an Einstein metric for the case that $$\rho <0.$$

Semiclassical Eigenfunctions of the Schrödinger Operator on a Graph That Are Localized Near a Subgraph

We study semiclassical eigenvalues and eigenfunctions of the Schrodinger operator on a geometric graph. We show that nontrivial boundary conditions at vertices lead to the existence of eigenfunctions, concentrated near a single vertex. We also construct semiclassical eigenfunctions, localized near edges and discuss general construction of spectral series which correspond to a general subgraph.

The Topological Structure of Conformally Flat Riemannian Manifolds

In this paper, we will prove vanishing and finiteness theorems for \(L^2\)-harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrodinger operators involving the squared norm of the traceless Ricci form. From these theorems and the work of Li–Tam, we can obtain some one-end and finite ends results for the locally conformally flat Riemannian manifold.

Sinc-regularized techniques to compute eigenvalues of schrödinger operators on L 2 ( I ) ⊕ ℂ 2 $L^{2}(I)\oplus \mathbb {C}^{2}$

We introduced two different sinc-regularized techniques to compute eigenvalues of Schrodinger operators defined on the Hilbert space $L^{2}(0,1)\oplus \mathbb {C}^{2}$ because of the appearance of eigenvalue parameter in the boundary conditions. Both techniques improve significantly the error estimates resulting from using the classical sampling method. Furthermore, it treats completely the obstacle that, for some potentials, integrals resulting from applying the sinc method cannot be explicitly computed. We derive a rigorous error analysis that takes into account both truncation and perturbation errors. Results are exhibited numerically and graphically with comparisons.

Elliptic PDEs on Compact Ricci Limit Spaces and Applications

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. We apply these to the study of second-order differential calculus on such limit spaces.

On the number of eigenvalues of the discrete one-dimensional Dirac operator with a complex potential

In this paper we define a one-dimensional discrete Dirac operator on $${\mathbb {Z}}$$ . We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. We also estimate the number of eigenvalues for the discrete Schrodinger operator with complex potential on $${\mathbb {Z}}$$ . That is we extend the result obtained by Hulko (Bull Math Sci, to appear) to the whole $${\mathbb {Z}}$$ .

Optimal Hardy inequalities for Schrödinger operators on graphs

For a given subcritical discrete Schrodinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ 1. Our results rely on a criticality theory for Schrodinger operators on general weighted graphs.

The Second-Order Riesz Transforms Related to Schrödinger Operators Acting on BMO Type Spaces on the Stratified Lie Group

Let \(\mathcal {L}=-{\Delta } + V\) be a Schrodinger operator on stratified Lie group G, where V is a nonnegative potential satisfying the suitable reverse Holder’s inequality. In this paper, we study the boundedness of the second-order Riesz transforms such as \(\mathcal {L}^{-1} \nabla ^{2}\) and \(\mathcal {L}^{-1}(-{\Delta })\) on the spaces of BMO type for weighted case. We generalized the known results to the weighted case and the case of the stratified Lie group. So our results are new, even in the case \(\mathbb {R}^{n}\), and they have some interest in its own right.

Scattering Theory for the Defocusing Fourth Order NLS with Potentials

Based on the endpoint Strichartz estimates for the fourth order Schrodinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schrodinger operator. J. Funct. Anal., 274, 605–658 (2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader, B.: The cubic fourth-order Schrodinger equation. J. Funct. Anal., 256, 2473–2517 (2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity $$1 + \frac{8}{n} < p < 1 + \frac{8}{{n - 4}}$$ in dimensions n ≥ 7.