Superlinear Schrödinger operators

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

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→∞.     

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 .     

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.     

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.     

Spectrum and pseudospectrum of non-self-adjoint Schrödinger operators with periodic coefficients

We consider the pseudospectrum of the non-self-adjoint operator $$\mathfrak{D} = - h^2 \frac{{d^2 }}{{dx^2 }} + iV(x)$$ , where V(x) is a periodic entire analytic function, real on the real axis, with a real period T. In this operator, h is treated as a small parameter. We show that the pseudospectrum of this operator is the closure of its numerical image, i.e., a half-strip in ?. In this case, the pseudoeigenfunctions, i.e., the functions ?(h, x) satisfying the condition $$\left\| {\mathfrak{D}\varphi - \lambda \varphi } \right\| = O(h^N ), \left\| \varphi \right\| = 1, N \in \mathbb{N}$$ , can be constructed explicitly. Thus, it turns out that the pseudospectrum of the operator under study is much wider than its spectrum.     

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.     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

On the spectral properties of discrete Schrödinger operators

We use the method of the conjugate operator to prove a limiting absorption principle and the absence of the singular continuous spectrum for discrete Schrödinger operators. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V decaying arbitrarily slowly to zero at infinity.     

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrödinger 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 Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger 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.     

The spectrum of skew-shift Schrödinger operators contains intervals

I prove that the spectrum of a skew-shift Schrödinger operator contains large intervals in the semi-classical regime. In the semi-classical limit, these intervals approach the range of the potential.     

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      

Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

Spectral analysis of a class of Schrödinger difference operators

Doklady Mathematics -     

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups. The text was submitted by the authors in English.     

Schrödinger operators with slowly decaying potentials

Several recent papers have obtained bounds on the distribution of eigenvalues of non-self-adjoint Schrödinger operators and resonances of self-adjoint operators. In this paper we describe two new methods of obtaining such bounds when the potential decays more slowly than previously permitted.