Spectral triangles of Schrödinger operators with complex potentials

Consider the Schrödinger operator with a complex-valued potential v of period Let and be the eigenvalues of L that are close to respectively, with periodic (for n even), antiperiodic (for n odd), and Dirichelet boundary conditions on [0,1], and let be the diameter of the spectral triangle with vertices We prove the following statement: If then v(x) is a Gevrey function, and moreover      

Spectral analysis of a class of Schrödinger difference operators

Doklady Mathematics -     

An improved criterion for absence of eigenvalues of one-dimensional Schrödinger operators

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. As an application we consider quasiperiodic measures as potentials. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral properties of Schrödinger operators on radial N-dimensional infinite trees

We study the discreteness of the spectrum of Schrödinger operators which are defined on a class of radial N-dimensional rooted trees of a finite or infinite volume, and are subject to a certain mixed boundary condition. We present a method to estimate their eigenvalues using operators on a one-dimensional tree. These operators are called width-weighted operators, since their coefficients depend on the section width or area of the N-dimensional tree. We show that the spectrum of the width-weighted operator tends to the spectrum of a one-dimensional limit operator as the sections width tends to zero. Moreover, the projections to the one-dimensional tree of eigenfunctions of the N-dimensional Laplace operator converge to the corresponding eigenfunctions of the one-dimensional limit operator.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

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Superlinear Schrödinger operators

We find nontrivial and ground state solutions for the nonlinear Schrödinger equation under conditions weaker than those previously assumed.     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Spectral instability for some Schr?dinger operators

Summary. We show that the condition numbers of isolated eigenvalues of typical non-self-adjoint differential operators such as the harmonic oscillator may be extremely large. We describe a stable procedure for computing the condition numbers for Schr?dinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential. Received October 9, 1998 / Revised version received September 13, 1999 / Published online 16 March 2000     

Periodic magnetic Schrödinger operators: Spectral gaps and tunneling effect

A periodic Schrödinger operator on a noncompact Riemannian manifold M such that H ¹(M, ?) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field, the existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system.     

Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of “sparse potentials,” suggested by the authors for the lattice \mathbbZ^d {\mathbb{Z}^d} , d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schr?dinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \mathbbZ² {\mathbb{Z}^2} , where D = 2. Bibliography: 13 titles.     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Eigenvalues of Schrödinger operators and Dirichlet-to-Neumann maps

Eigenvalues and eigenspaces of selfadjoint Schrödinger operators on are expressed in terms of Dirichlet-to-Neumann maps corresponding to Schrödinger operators on the upper and lower half space. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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...