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.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Universal spaces of subdifferentials of sublinear operators ranging in the cone of bounded lower semicontinuous functions

We study Fréchet’s problem of the universal space for the subdifferentials ?P of continuous sublinear operators P: V → BC(X)_～ which are defined on separable Banach spaces V and range in the cone BC(X)_～ of bounded lower semicontinuous functions on a normal topological space X. We prove that the space of linear compact operators L ^c(? ², C(βX)) is universal in the topology of simple convergence. Here ? ² is a separable Hilbert space, and βX is the Stone-?ech compactification of X. We show that the images of subdifferentials are also subdifferentials of sublinear operators.     

A note on list improper coloring of plane graphs

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v∈V(G). An (L,d)^∗-coloring is a mapping ? that assigns a color ?(v)∈L(v) to each vertex v∈V(G) such that at most d neighbors of v receive color ?(v). A graph is called (k,d)^∗-choosable, if G admits an (L,d)^∗-coloring for every list assignment L with |L(v)|≥k for all v∈V(G). In this note, it is proved that every plane graph, which contains no 4-cycles and l-cycles for some l∈{8,9}, is (3,1)^∗-choosable.     

Riesz Transforms of Schr?dinger Operators on Manifolds

We consider Schr?dinger operators A=???+V on L ^p (M) where M is a complete Riemannian manifold of homogeneous type and V=V ⁺?V ^? is a signed potential. We study boundedness of Riesz transform type operators $\nabla A^{-\frac{1}{2}}$ and $|V|^{\frac{1}{2}}A^{-\frac{1}{2}}$ on L ^p (M). When V ^? is strongly subcritical with constant ????(0,1) we prove that such operators are bounded on L ^p (M) for $p\in(p_{0}', 2]$ where $p_{0}'=1$ if N??2, and $p_{0}'=(\frac{2N}{(N-2)(1-\sqrt{1-\alpha })})' \in (1, 2)$ if N>2. We also study the case p>2. With additional conditions on V and M we obtain boundedness of ?A ^?1/2 and |V|^1/2 A ^?1/2 on L ^p (M) for p??(1,inf?(q ₁,N)) where q ₁ is such that $\nabla(-\Delta)^{-\frac{1}{2}}$ is bounded on L ^r (M) for r??[2,q ₁).     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.     

Spectral analysis of a Dirac operator with a meromorphic potential

We consider an operator Q(V) of Dirac type with a meromorphic potential given in terms of a function V of the form V(z)=λV₁(z)+μV₂(z), z∈C?{0}, where V₁ is a complex polynomial of 1/z, V₂ is a polynomial of z, and λ and μ are nonzero complex parameters. The operator Q(V) acts in the Hilbert space L²(R²;C⁴)=^⊕4L²(R²). The main results we prove include: (i) the (essential) self-adjointness of Q(V); (ii) the pure discreteness of the spectrum of Q(V); (iii) if V₁(z)=z−p and 4?degV₂?p+2, then kerQ(V)≠{0} and dimkerQ(V) is independent of (λ,μ) and lower order terms of ∂V₂/∂z; (iv) a trace formula for dimkerQ(V).     

On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrödinger-type differential expressions L(κ)=D²+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=^*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L²(E) and L²(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.     

Perturbed sampling formulas and local reconstruction in shift invariant spaces

Let V? be a closed subspace of L₂(R) generated from the integer shifts of a continuous function ? with a certain decay at infinity and a non-vanishing property for the function Φ^†(γ)=_∑n∈Z?(n)e−inγ on [−π,π]. In this paper we determine a positive number δ? so that the set {n+δn}_n∈Z is a set of stable sampling for the space V? for any selection of the elements δn within the ranges ±δ?. We demonstrate the resulting sampling formula (called perturbation formula) for functions f∈V? and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well.     

On the absolute continuity of Schrödinger operators with spherically symmetric,long-range potentials,I

Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = V_S(r) + V_L(r). Assume that for some r₀, V_L(r) ?C^2k(r₀, ∞) and that there exist μ > 0, δ > 0, such that $\text{(}\text{d}\text{dr}\text{)}{}^{\mathrm{j}}\text{V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) = O(r}{}^{\mathrm{?\mu ?j\delta }}\text{)}\text{as}\text{r → ∞, 1 ? j ? 2k}$. Assume further that min(2kμ, (2k ? 1)δ + μ) > 1. Under this weak decay condition on V_L(r) it is shown in this paper that the positive spectrum of H is absolutely continuous and that the absolutely continuous part of H is unitarily equivalent to ?Δ, provided that the singularity of V at 0 is properly restricted. In particular, some oscillation of V_L(r) at infinity is allowed.     

On the double commutant of Cowen-Douglas operators

Let T be a Cowen-Douglas operator. In this paper, we study the von Neumann algebra V^?(T) consisting of operators commuting with both T and T^? from a geometric viewpoint. We identify operators in V^?(T) with connection-preserving bundle maps on E(T), the holomorphic Hermitian vector bundle associated to T. By studying such bundle maps, the structure of V^?(T) as well as information on reducing subspaces of T can be determined.     

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.     

On the absolute continuity of Schrödinger operators with spherically symmetric,long-range potentials,II

Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = V_S(r) + V_L. Let λ = lim sup_r→∞V_L(r) < ∞ (we allow λ = ? ∞) and set λ⁺ = max(λ, 0). Assume that for some r₀, V_L(r) ?C^2k(r₀, ∞) and that there exists δ > 0 such that $\text{(}\text{d}\text{dr}\text{)}{}^{\mathrm{j}}\text{V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) · (λ}{}^{\mathrm{+}}\text{? V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) + 1)}{}^{\mathrm{?1}}\text{= O(r}{}^{\mathrm{?j\delta }}\text{), j = 1,…, 2k,}\text{as}\text{r → ∞}$. Assume further that $\text{∝}{}_1{}^{\mathrm{\infty }}\text{(}\text{dr}\text{¦ V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r)¦}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{) = ∞}$ and that 2kδ > 1. It is shown that: (a) The restriction of H to C^∞(Rⁿ) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous.     

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.     

Large time behavior of the Lp norm of Schrödinger semigroups

LetV ? 0, V?C₀^∞(R^v) with v ? 3 be such that $\text{H = ?}\text{1}\text{2}\text{Δ + V ? 0}$ but for any $\text{ε > 0, ?}\text{1}\text{2}\text{Δ + (1 + ε)V}$ is not positive. We determine the exact rate of divergence of the norm of e^?tH as a map from L^∞ to L^∞. A number of related problems are discussed.     

A perturbation theorem for operator semigroups in Hilbert spaces

We prove a perturbation result for C ₀-semigroups on Hilbert spaces and use it to show that certain operators of the form Au = iu ^(2k) + V · u ^(l) on L ² (?) generate a semigroup that is strongly continuous on (0, ∞).     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.     

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?).