Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

Microlocal properties of scattering matrices

We consider the scattering theory for a pair of operators H₀ and H = H₀ + V on L²(M, m), where M is a Riemannian manifold, H₀ is a multiplication operator on M, and V is a pseudodifferential operator of order ? μ, μ > 1. We show that a time-dependent scattering theory can be constructed, and the scattering matrix is a pseudodifferential operator on each energy surface. Moreover, the principal symbol of the scattering matrix is given by a Born approximation type function. The main motivation of the study comes from applications to discrete Schrödigner operators, and it also applies to various differential operators with constant coefficients and short-range perturbations on Euclidean spaces.     

The semiclassical resolvent and the propagator for non-trapping scattering metrics

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M^○,g). (Euclidean Rn, with a compactly supported metric perturbation, is an example of such a space.) Let Δ be the positive Laplacian on (M,g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator ⁻¹(h²Δ+V−²(λ₀±i0)), at a non-trapping energy λ₀>0, uniformly for h∈(0,h₀), h₀>0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e−it(Δ/2+V), t∈(0,t₀) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.     

Spectral analysis of nonselfadjoint Schrödinger operators with a matrix potential

Dissipative Schrödinger operators with a matrix potential are studied in L₂((0,∞);E)(dimE=n<∞) which are extension of a minimal symmetric operator L₀ with defect index (n,n). A selfadjoint dilation of a dissipative operator is constructed, using the Lax-Phillips scattering theory, the spectral analysis of a dilation is carried out, and the scattering matrix of a dilation is founded. A functional model of the dissipative operator is constructed and its characteristic function's analytic properties are determined, theorems on the completeness of eigenvectors and associated vectors of a dissipative Schrödinger operator are proved.     

Analytic scattering theory of two-body Schrödinger operators

We consider the self-adjoint analytic family of operators H(z) in L²(R^m) defined for $\text{z ? S}{}_{\mathrm{\alpha }}\text{= {z ∥}\text{Arg}\text{z ¦ < α}}$, associated with the operator H = H(1) = H₀ + V, where H₀ = ?Δ and V is a dilation-analytic short-range potential. The analytic connection between the local wave and scattering operators associated with the operators H(e^i?) is established. The scattering matrix S(?) of H has a meromorphic continuation S(z) to S_α with poles precisely at the resolvent resonances of H, and the local scattering operators of e^?2i?H(e^i?) have representations in terms of the analytically continued scattering matrix S(?e^i?).     

Self-adjointness of Schrödinger-type operators with locally integrable potentials on manifolds of bounded geometry

We consider a Schrödinger-type differential expression , where ∇ is a C^∞-bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry (M,g) with metric g and positive C^∞-bounded measure dμ, and V∈L_loc¹(EndE) is a linear self-adjoint bundle map. We define the maximal operator H_V,max associated to H_V as an operator in L²(E) given by H_V,maxu=H_Vu for all , where ∇∗∇u in is understood in distributional sense. We give a sufficient condition for the self-adjointness of H_V,max. The proof adopts Kato's technique to our setting, but it requires a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of u∈L²(M) satisfying the equation (Δ_M+b)u=ν, where Δ_M is the scalar Laplacian on M, b>0 is a constant and ν?0 is a positive distribution on M. For local estimates, we use a family of cut-off functions constructed with the help of regularized distance on manifolds of bounded geometry.     

Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces

Let H be a separable complex Hilbert space, A a von Neumann algebra in ?(H),a faithful, normal state on A. We prove that if a sequence (X_n: n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σ^∞_n=1n⁻² Var(X_n) log²(n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L₂=L₂(A,φ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.     

A Generalization of the Direct and Inverse Problem for the Radial Schrödinger Equation

We consider the operators H₀= ?d²/dr² and H₁ = ?d²/dr² + V(r) (0< r< ∞) acting on a Hilbert space of complex functions f(r) such that the subspaces in which the operators are defined consist of twice differentiable functions which satisfy the boundary condition (d/dr)f(0) = αf(0). H₁ and H₀ are Hermitian in this subspace. Assuming V(r)→0 as r→∞ sufficiently rapidly, the scattering operator formalism is set up for the direct scattering problem. Next we consider the inverse problem of determining V(r) from H₀ and the spectral measure function for the spectrum of H₁ through the use of an appropriate Gelfand-Levitan equation. It is shown that generally the value of α associated with H₁ differs from that for H₀, i.e., H₁ and H₀ generally operate in different subspaces. Thus scattering cannot be defined. However, by changing the spectral measure function, one obtains a new Gelfand-Levitan equation such that H₁ is the same as before [i.e., α and V(r) are the same] from the operator H₀, which uses the same value of α as H₁. Thus H₁ and the new H₀ operate in the same subspace of Hilbert space, and scattering can be defined. The process of obtaining the new H₀ after finding H₁ from the old H₀ is somewhat analogous to renormalization in field theory, where a new H₀ is picked to have properties compatible with H₁. A necessary and sufficient condition on the spectral data is given which makes the domains of H₀ and H₁ coincide and thus makes “renormalization” unnecessary. The direct problem is a generalization of the usual l=0 radial Schrödinger equation. The inverse problem is a generalization of the corresponding inverse problem. It is also a generalization of the case α=0 for H₀ considered by Gelfand and Levitan in their early work on the inverse spectral problem. An incompletely understood connection of the inverse problem for the radial equation to solutions of the Korteweg-deVries equation in the half space is discussed. The existence of such a connection is one of the motivations for studying the generalized radial Schrodinger equation.     

L boundedness of commutator operator associated with schrödinger operators on heisenberg group

Let L = −ΔHⁿ + V be a Schrödinger operator on Heisenberg group Hⁿ, where ΔHⁿ is the sublaplacian and the nonnegative potential V belongs to the reverse Hölder class B_Q/2, where Q is the homogeneous dimension of Hⁿ  . Let T₁=(−ΔHn+V)⁻¹V,T₂=(−ΔHn+V)^−1/V₂1/2$T_1={(-\Delta H^n+V)}^{-1}V,T_2={(-\Delta H^n+V)}^{-1/2_{V}1/2}$, and T₃=(−Δ_Hn+V)^−1/2_∇Hn$T_3={(-{\Delta }_{H^n}+V)}^{-1/2}{\nabla }_{H^n}$, then we verify that [b, T_i], i = 1,2,3 are bounded on some L^p(Hⁿ), where b ∈ BMO(Hⁿ). Note that the kernel of T_i, i = 1,2,3 has no smoothness.     

A limiting absorption principle for Schrödinger operators with spherically symmetric exploding potentials

LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V _S(r)+V _L(r), whereV _S(r) is short-range and the exploding partV _L(r) satisfies the following assumptions: (a) Λ=lim sup _r→∞ V _L(r)<∞ (but Λ=?∞ is possible). Denote Λ⁺= max(Λ,0). (b)V _L(r)∈C ^2k (r ₀, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) ^j V _L(r) · (Λ⁺?V _L(r))^?1=O(r ^jδ) asr → ∞,j=1, ..., 2k. (c) ∫ _r0 ^∞ dr|V _L(r|^1/2 dr|V _L(r)|^1/2=∞. (d) (d/dr)V _L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)^?1 is established. More specifically, ifK ?C ⁺={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L ²(R ⁿ) is a weighted-L ² space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced.     

Spectral shift function of higher order

This paper resolves affirmatively Koplienko’s (Sib. Mat. Zh. 25:62–71, 1984) conjecture on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions. A spectral shift function of order n∈? is the function η _n =η _n,H,V such that $$ \operatorname {Tr}\Biggl( f(H + V)-\sum_{k = 0}^{n-1} \frac{1}{k!}\, \frac{d^k}{dt^k} \bigl[ f(H + tV) \bigr] \bigg|_{t = 0} \Biggr) = \int_\mathbb{R}f^{(n)} (t)\, \eta_n (t)\, dt, $$ for every sufficiently smooth function f, where H is a self-adjoint operator defined in a separable Hilbert space ? and V is a self-adjoint operator in the n-th Schatten-von Neumann ideal S ⁿ . Existence and summability of η ₁ and η ₂ were established by Krein (Mat. Sb. 33:597–626, 1953) and Koplienko (Sib. Mat. Zh. 25:62–71, 1984), respectively, whereas for n>2 the problem was unresolved. We show that η _n,H,V exists, integrable, and $$\Vert \eta_n \Vert _{L^1(\mathbb{R})} \leq c_n \Vert V \Vert _{S^n}^n, $$ for some constant c _n depending only on n∈?. Our results for η _n rely on estimates for multiple operator integrals obtained in this paper. Our method also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.     

Pointwise convergence of eigenfunction expansions associated with ordinary differential operators

With an ordinary differential expression L = ∑ⁿ_k=0P_kD^k on an open interval I?$\text{r}$ is associated a selfadjoint operator H in a Hilbert space, possibly beyond $\text{K}$=L²(l). The set $\text{D}$H∩$\text{K}$ only depends on the generalized spectral family associated with H. It is shown that the (differentiated) eigenfunction expansion given by H converges uniformly on compact subintervals of l for functions in $\text{D}$(H)∩$\text{L}$ In case H is a semibounded selfadjoint operator in $\text{K}$=L²T, a similar result is proved for functions in $\text{D}$|H|, which is the set of all $\text{K}$∈$\text{K}$ for which there exists a sequence f_n∈(H) such that f_n→f in $\text{H}$ and (H(f_n ? f_m), f_n ? f_m → 0 as n, m → ∞.     

On the density of states for the periodic Schrödinger operator

An asymptotic formula for the density of states of the polyharmonic periodic operator (?δ) ^l +V inR ⁿ ,n≥2,l>1/2 is obtained. Special consideration is given to the case of the Schrödinger equationn=3,l=1,V being a periodic potential, where the second term of the asymptotic is found.     

Resolvent estimates for a certain class of Schrödinger operators with exploding potentials

Let $\text{H = ?Δ + V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)+ V(x)}$ be a Schrödinger operator in Rⁿ. Here $\text{V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)}$ is an “exploding” radially symmetric potential which is at least C² monotone nonincreasing and O(r²) as r → ∞. V is a general potential which is short range with respect to V_E. In particular, V_E  0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = lim_{r → ∞}V_E(r) and R(z) = (H ? z)^?1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset $\text{N}$?(Λ, ∞), in a suitable operator topology $\text{B(L, L}{}^1\text{)}$. And L ? L²(Rⁿ) is a weighted L²-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing.     

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H _m+H_om+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H _m, H _om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.     

Inverse scattering problem for nonstationary Dirac-type systems on the half-plane

In this paper was considered the scattering problem for the nonstationary Dirac-type systems of n (n?2) equations on the half-plane when the system has n₁ (1?n₁?n−1) incident and n₂ (n₂=n−n₁) scattered waves. In case n₁ is divisible by n₂, we formulate the inverse scattering problem for a nonstationary Dirac-type system when considering m () scattering problems on the half-plane with the same incident waves but different boundary conditions. Moreover, the scattering operator for the nonstationary Dirac-type system on half-plane was defined and unique restoration of the potential with respect to the scattering operator was proved.     

Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

On the localization of binding for Schrödinger operators and its extension to elliptic operators

In this paper we study the asymptotic behavior of the ground state energyE(R) of the Schrödinger operatorP _R=?Δ+V ₁(x)+V ₂(x-R),x,R ∈?ⁿ, where the potentialsV _i are small perturbations of the Laplacian in ?ⁿ,n≥3. The methods presented here apply also in the investigation of the ground state energyE(g) of the operatorPg=P+V ₁(x)+V ₂(gx), x ∈X,g ∈G, whereP _g is an elliptic operator which is defined on a noncompact manifoldX, G is a discrete group acting onX by diffeomorphismsG×X∈(g, x)→gx∈X, andP is aG-invariant elliptic operator which is subcritical inX.     

Regularity properties of Schrödinger and Dirichlet semigroups

First we compute Brownian motion expectations of some Kac's functionals. This allows a complete study of the semigroups generated by the formal differential operator $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ on the various Lebesgue's spaces L^q=L^q$\text{R}$ⁿ, dx, whenever the negative part of V is in L^∞ + L^p for some $\text{p >}\text{max}\text{{1,}\text{n}\text{2}\text{}}$. Our approach is probabilistic and some of the proofs are surprisingly elementary. The negative infinitesimal generators of our semigroups are shown to be reasonable self-adjoint extensions of H. Under mild assumptions on V, H is unitary equivalent to the Dirichlet operator, say D, associated to its groundstate measure. We study regularity of the semigroups generated by D. We concentrate on hyper and supercontractivity and we give, using probabilistic techniques, new examples of potential functions V which give rise to hyper and supercontractive Dirichlet semigroups.