Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.     

The resolvent of a dilation-analytic three-particle system

If the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λ_p, φ) starting at the thresholds λ_p of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable $\text{L}$^p-spaces, each half-line Y(λ_p, φ) is associated with an operator P(λ_p, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λ_p, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λ_p when φ runs through some interval $\text{0 < α ? φ ? β <}\text{π}\text{2}$. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λ_p, φ) that are sufficient for P(λ_p, φ)[H(φ) ? λ_p] e^?2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λ_p + ze^2iφ, the resolvent R(λ_p + ze^2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λ_p, φ) is included.     

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),

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

.     

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?).     

A fundamental region for Hecke's modular group

Hecke proved analytically that when λ ≥ 2 or when $\text{λ = 2}\text{cos}\text{(}\text{π}\text{q}\text{)}$, q ∈ Z, q ≥ 3, then $\text{B(λ) = {τ:}\text{Im}\text{τ > 0, |}\text{Re}\text{τ| <}\text{λ}\text{2}\text{, |τ| > 1}}$ is a fundamental region for the group G(λ) = 〈S_λ, T〉, where S_λ: τ → τ + λ and $\text{T: τ →}\text{?1}\text{τ}$. He also showed that B(λ) fails to be a fundamental region for all other λ > 0 by proving that G(λ) is not discontinuous. We give an elementary proof of these facts and prove a related result concerning the distribution of G(λ)-equivalent points.     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Small diameter interchange graphs of classes of matrices of zeros and ones

Let $\text{A}$(R, S) denote the class of all m×n matrices of 0's and 1's having row sum vector R and column sum vector S. The interchange graph G(R, S) is the graph where the vertices are the matrices in $\text{A}$(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We characterize those $\text{A}$(R, S) for which the graph G(R, S) has diameter at most 2 and those $\text{A}$(R, S) for which G(R, S) is bipartite.     

On the maximum number of permutations with given maximal or minimal distance

Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number $\text{M}$ such that there exist $\text{M}$ different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(p^m, ? 2) = (p^m ? 2)!, R(p^m + 1, ? 3) = (p^m ? 2)!, $\text{R(k, ? k ? 3) =}\text{k!}\text{2}$, R(k, ? 0) = k, R(p^m, ? 1) = p^m(p^m ? 1), R(p^m + 1, ? 2) = (p^m + 1)p^m(p^m ? 1). The exact value of R(k, ? λ) is determined whenever k ? k₀(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k₀(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for $\text{λ = (}\text{k}\text{2}\text{) + O(}\text{k}\text{log}\text{k}\text{)}$.     

The chromatic polynomial of an unlabeled graph

We investigate the chromatic polynomial χ(G, λ) of an unlabeled graph G. It is shown that $\text{χ(G, λ) = (}\text{1}\text{|A(g)|}\text{) Σ}{}_{\mathrm{\pi \; \in \; A(g)}}\text{χ(g, π, λ)}$, where g is any labeled version of G, A(g) is the automorphism group of g and χ(g, π, λ) is the chromatic polynomial for colorings of g fixed by π. The above expression shows that χ(G, λ) is a rational polynomial of degree n = |V(G)| with leading coefficient $\text{1}\text{|A(g)|}$. Though χ(G, λ) does not satisfy chromatic reduction, each polynomial χ(g, π, λ) does, thus yielding a simple method for computing χ(G, λ). We also show that the number N(G) of acyclic orientations of G is related to the argument λ = ?1 by the formula $\text{N(G) = (}\text{1}\text{|A(g)|}\text{) Σ}{}_{\mathrm{\pi \; \in \; A(g)}}\text{(?1)}{}^{\mathrm{s(\pi )}}\text{χ(g, π, ?1)}$, where s(π) is the number of cycles of π. This information is used to derive Robinson's (“Combinatorial Mathematics V” (Proc. 5th Austral. Conf. 1976), Lecture Notes in Math. Vol. 622, pp. 28–43, Springer-Verlag, New York/Berlin, 1977) cycle index sum equations for counting unlabeled acyclic digraphs.     

Diagonal norm hermitian matrices

If v is a norm on $\text{C}$ⁿ, let H(v) denote the set of all norm-Hermitians in $\text{C}$ⁿⁿ. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ₁,…, λ_r, r?n, there is a norm v such that h ∈ H(v), but h^s?H(v), for some integer s, if and only if λ₂–λ₁,…, λ_r–λ₁ are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.     

On the uniqueness of embedding a residual design

We prove that if a residual $\text{2-(k(k+λ?1)}\text{λ,k,λ)}$ design R has more than one embedding into a symmetric design then k ? λ(λ?1)². If equality holds then R has exactly two embeddings and the corresponding derived design is in both cases λ ? 1 identical copies of the design of points and lines of PG(3, λ ? 1). Using the main proposition from which these results follow we also prove that if a symmetric2-(v,k, λ) design has an axial non-central or central non-axial automorphism then k?λ(λ² ? 2λ + 2).     

Qualitative behavior of special functions on compact symmetric spaces

For a symmetric space $\text{G}\text{K}$ of compact type, the highest-weight vectors for representations of G occurring in $\text{L}{}^2\text{(G}\text{K)}$ become heavily concentrated near certain submanifolds of $\text{G}\text{K}$ as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = P_λqP_λ, where $\text{P}{}_{\mathrm{\lambda }}\text{: L}{}^2\text{(}\text{G}\text{K}\text{) → V}{}_{\mathrm{\lambda }}$ is an orthogonal projection onto a G-irreducible summand, and q: G/K → $\text{R}$ is a continuous function acting on $\text{L}{}^2\text{(G}\text{K)}$ by multiplication.     

The number of Hamiltonian circuits

G = 〈V(G), E(G)〉 denotes a directed graph without loops and multiple arrows. $\text{K}$(G) denotes the set of all Hamiltonian circuits of G. Put H(n, r) = max{|E(G)|, |V(G)| = n, 1 ≤ |$\text{K}$(G)| ≤ r}. Theorem: $\text{H(n, 1) = (}\text{n}{}^2\text{2}\text{) + (}\text{n}\text{2}\text{) ?1}$. Further, H(n, 2),…, H(n, 5) are given.     

Connected matroids with the smallest whitney numbers

In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if _X(G) = Σμ(0, x)λ^{r ? r(x)} = Σ (?1)^{r ? k}W_kλ^k is the characteristic polynomial of G, then

$\text{w}{}_{\mathrm{k}}\text{?}\text{r}\text{k}\text{+n}\text{r?1}\text{k}$

Thus γ(G) ? 2^{r ? 1} (n+2), where γ(G) = Σw_k. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then

$\text{w}{}_{\mathrm{i}}\text{?}\text{r}\text{i}\text{+ n}\text{r}\text{i+1}\text{for i>1; w}{}_1\text{?r+n}\text{r}\text{2}\text{? 1;}$

|μ| ? (r? 1)n; and γ ? (2^{r ? 1} ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, $\text{r(G) = r(}\text{G}\text{?}\text{) = 4}$, and $\text{n(G) = n(}\text{G}\text{?}\text{) = 3}$ then $\text{(w}{}_1\text{(G))}{}^4{}_{\mathrm{t-G}}\text{? (w}{}_1\text{(}\text{G}\text{?}\text{)) = (8, 20, 18, 7, 1)}$. Further, if β is the Crapo invariant,

$\text{β(G)=}\text{d}{}_{\mathrm{X}}\text{(G)}\text{dλ}\text{(1)}\text{,}$

then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network.     

Duality theorems for perturbed convex optimization

We show that if F, X are two locally convex spaces and $\text{h: F →}\text{R}\text{?}\text{, ?: F × X → R}$ are two convex functionals satisfying h(y) = ?(y, x₀) (y?F) for some x₀?X, then, under suitable assumptions, the computation of inf h(F) can be reduced to the computation of inf ?(H) on certain hyperplanes H of F × X. We give some applications.     

Gaussian processes with biconvex covariances

Let R(s, t) be a continuous, nonnegative, real valued function on a ≤ s ≤ t ≤ b. Suppose $\text{?R}\text{?s}\text{≥ 0}$, $\text{?R}\text{?t}\text{≤ 0}$, and $\text{?}{}^2\text{R}\text{?t}\text{?t ≤ 0}$ in the interior of the domain. Then the extension of R to a symmetric function on [a, b] × [a, b] is a covariance function. Such a covariance is called biconvex. Let X(t) be a Gaussian process with mean 0 and biconvex covariance. X has a representation as a sum of simple moving averages of white noises on the line and plane. The germ field of X at every point t is generated by X(t) alone. X is locally nondeterministic. Under an additional assumption involving the partial derivatives of R near the diagonal, the local time of the sample function exists and is jointly continuous almost surely, so that the sample function is nowhere differentiable.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

Bifurcation for a class of singular elliptic problems with quadratic convection term

We study the bifurcation problem ?Δu=g(u)+λ|?u|²+μ in $\text{Ω,}\text{u=0}$ on $\text{?Ω}$, where λ,μ?0 and $\text{Ω}$ is a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ₁, where a=lim_t→+∞g(t) and λ₁ is the first eigenvalue of the Laplace operator in $\text{H}{}^1{}_0\text{(Ω)}$. We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On spectral characterizations and embeddings of graphs

The least eigenvalue of the 0-1 adjacency matrix of a graph is denoted λ G. In this paper all graphs with λ(G) greater than ?2 are characterized. Such a graph is a generalized line graph of the form L(T;1,0,…,0), L(T), L(H), where T is a tree and H is unicyclic with an odd cycle, or is one of 573 graphs that arise from the root system E₈. If G is regular with λ(G)>?2, then Gis a clique or an odd circuit. These characterizations are used for embedding problems; λ_R(H) = sup{λ(G)z.sfnc;HinG; Gregular}. H is an odd circuit, a path, or a complete graph iff λ_R(H)> ?2. For any other line graph H, λ_R(H) = ?2. A similar result holds for complete multipartite graphs.