Calculation of the eigenvalues of Schrödinger equations by an extension of Hill's method

The eigenfunctions of the one dimensional Schrödinger equation Ψ″ + [E ? V(x)]Ψ=0, where V(x) is a polynomial, are represented by expansions of the form $\text{∑}\text{k}\text{=0}\text{∞}{}^{\mathrm{<mtext>c</mtext>}}\text{k}{}^{\mathrm{?}}\text{k}\text{(ω,}\text{x}\text{)}$. The functions ?_k (ω, x) are chosen in such a way that recurrence relations hold for the coefficients c_k: examples treated are D_k(ωx) (Weber-Hermite functions), exp (?ωx²)x^k, exp (?cx^q)D_k(ωx). From these recurrence relations, one considers an infinite bandmatrix whose finite square sections permit to solve approximately the original eigenproblem. It is then shown how a good choice of the parameter ω may reduce dramatically the complexity of the computations, by a theoretical study of the relation holding between the error on an eigenvalue, the order of the matrix, and the value of ω. The paper contains tables with 10 significant figures of the 30 first eigenvalues corresponding to V(x) = x^2m, m = 2(1)7, and the 6 first eigenvalues corresponding to V(x) = x² + λx¹⁰ and x² + λx¹², λ = .01(.01).1(.1)1(1)10(10)100.     

A necessary condition for the local solvability of the operator Pm2(x, D) + P2m − 1(x, D)

It is shown that a necessary condition for the local solvability of the operator P(x, D) = P_m²(x, D) + P_{2m ? 1}(x, D), where P_m(x, D) is an mth-order homogeneous differential operator of principal type with real coefficients, is that along any null-bicharacteristic strip of P_m(x, ξ) the imaginary part of the sub-principal symbol cannot have an odd-order zero where its real part does not vanish.     

The critical Fujita exponent for a diffusion equation with a potential term

We study the initial-boundary-value problem of the diffusion equation u _t = Δu ^m ? V (x)u ^m + u ^p in a conelike domain D = [1,∞) × Ω, where V (x) ~ ω ₂ |x| ^?2 with ω ₂ > 0. Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω, and let l denote the positive root of l ² + (n ? 2)l = ω ₁ + ω ₂. We prove that if m < p ≤ m + 2/(n + l), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if p > m + 2/(n + l), then the problem has global solutions for some \( {u_0}\gneq 0 \) .     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

On multiple ergodicity of affine cocycles over irrational rotations

Let \({T_\alpha }\) denote the rotation \({T_\alpha }x = x + \alpha \) (mod 1) by an irrational number α on the additive circle T = [0, 1). Let β ₁, …, β _d be d ≥ 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in ? ^d+1) generated over \({T_\alpha }\) by the vectorial function Ψ _d+1(x):= (φ(x), φ(x+β ₁), …, φ(x+β _d )), with φ(x) = {x}?½. It was already proved in [LeMeNa03] that Ψ₂ is regular for α with bounded partial quotients. In the present paper we show that Ψ₂ is regular for any irrational α. For higher dimensions, we give sufficient conditions for regularity. While the case d = 2 remains unsolved, for d = 3 we provide examples of non-regular cocycles Ψ₄ for certain values of the parameters β ₁, β ₂, β ₃. We also show that the problem of regularity for the cocycle Ψ _d+1 reduces to the regularity of the cocycles of the form \({\Phi _d} = {({1_{[0,{\beta _j}]}} - {\beta _j})_{j = 1, \ldots ,d}}\) (taking values in ? ^d ). Therefore, a large part of the paper is devoted to the classification problems of step functions with values in ? ^d .     

Spectral asymptotics of a nonself-adjoint elliptic differential operator with an indefinite weight function

In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R ⁿ , where A is an elliptic differential operator of order 2m with domain D(A) ? W _m ^2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure.     

The Reconstruction of the Attracting Potential in the Sturm-Liouville Equation through Characteristics of Negative Discrete Spectrum

Let us consider the Sturm-Liouville equation on the positive half-axis with negative potential of the form q(x) = ω²Q(x)+Q₀(x), where functions Q and Q₀ are integrable together with derivatives of the order m + 1 and have polynomial decreasing at infinity. In the development of the Lax-Levermore result we show that the function Q(x) + ω^?2Q₀(x) can be reconstructed with accuracy O(ω^?m)( only through characteristics of discrete negative spectrum of the Dirichlet problem for(*). As an application we prove that it is possible to reconstruct with prescribed accuracy a density and a compressibility of the horizontal homogeneous liquid half-space through wavenumbers and amplitudes of surface waves excited by monochromatic source with sufficiently large but fixed frequencies ω₁ and ω₂.     

Computations of coefficients in the polynomials of Pade´approximations by solving systems of linear equations

We explore reliability, stability and accuracy of determining the polynomials which define the Pade´approximation to a given function h(x) by solving a system of linear equations to get the coefficients in the denominator polynomial B_n(x). The coefficients in the numerator polynomial A_m(x) follow directly from those for B_n(x). Our approach is in the main heuristic. For the numerics we use the models e^?x1n(1 +x), (1 +x)^{± 1/2} and the exponential integral, each with m=n. The system of equations, with matrix of Toeplitz type, was solved by Gaussian elimination (Crout algorithm) with equilibration and partial pivoting. For each model, the maximum number of incorrect figures in the coefficients is of the order n at least, thus indicating that the matrix becomes ill conditioned as n increases. Let δ_n(x)andω_n(x) be the errors in A_n(x) and B_n(x) respectively, due to errors in the coefficients of B_n(x). For x fixed, δ_n(x) and ω_n(x) and the corresponding relative errors increase as n increases. However, for a considerable range on n, the relative errors in A_n(x)B_n(x) are virtually nil. This has the following theoretical explanation. Now B_n(x)h(x) ?A_m(m) = 0 (x^{m+n+ 1}). It can be shown that ω_n(x)h(x) ? δ_m(x) = 0(x^{m+ 1}). In this sense both A_m(x)B_n(x)andδ_m(x)ω_n(x) are approximations to h(x). Thus if the difference of these two approximations and ω_n(x)B_n(x), the relative error in B_n(x), are sufficiently small, then the relative error in A_m(x)/B_n(x) is of no consequence.     

On the growth order of the rectangular partial sums of multiple non-orthogonal series

Пустьd-натуральное ч исло,Z ^d — множество на боров k=(k ₁, ...,k _d ), состоящих из неотрицательных цел ыхk _j ,Z ₊ ^d =k∈Z ^d :k≧1. Предположи м, что системаf _k (x):k∈Z ₊ ^d ? ?L₂(X,A, μ) и последовател ьностьa _k :k∈Z ₊ ^d . таковы, чт о для всех b∈Z^d и m∈Z ₊ ^d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z ₊ ^d положительн а и не убывает. Например, есл иf _k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {a_k}. В работе получены оце нки порядка роста пря моугольных частных суммS _m (x)= =∑ a_kf_k(x) при maxmj→∞ как в случ ае {a_k}∈l₂, таки для {a_k}l₂. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.     

Equivalence of measures for some class of Gaussian random fields

We consider two Gaussian measures P₁ and P₂ on (C(G), $\text{B}$) with zero expectations and covariance functions R₁(x, y) and R₂(x, y) respectively, where R_ν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A^(ν) of order $\text{2m, m ≥ [}\text{d}\text{2}\text{] + 1}$, on a bounded domain G in $\text{R}$^d (ν = 1, 2). It is shown that if the order of A⁽²⁾ ? A⁽¹⁾ is at most $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are equivalent, while if the order is greater than $\text{2m ? [}\text{d}\text{2}\text{] ? 1}$, then P₁ and P₂ are not always equivalent.     

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions dφ₁(x) and d₂(x) such that dφ₂(x) = (1 + kx²)d₁(x). As applications of properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

A Wiener test for integrals of brownian motion and the existence of smooth curves in nowhere dense sets

Let J_ω^x(t) = x + ∝₀^tb_ω(s) ds, where b_ω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process J_ω^x(t): Let K be a compact plane set and let x?K. Then if ∑ 2ⁿM₁(A_n(x)?K) < ∞ (where $\text{A}{}_{\mathrm{n}}\text{(x) = {2}{}^{\mathrm{?n?1}}\text{? ¦ z ? x ¦ ? 2}{}^{\mathrm{?n}}\text{}}$ and M₁ denotes one-dimensional Hausdorff content), the process J_ω^x(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes.     

Domains with growth conditions for the quasihyperbolic metric

For a given growth functionH, we say that a domainD ?R ⁿ is anH-domain if δ_D x≤δ_D(x ₀)H(k _D(x,x ₀)),x ∈D, where δ_D(x)=d(x?D) andk _D denotes the quasihyperbolic distance. We show that ifH satisfiesH(0)=1, |H'|≤H, andH"≤H, then there exists an extremalH-domain. Using this fact, we investigate some fundamental properties ofH-domains.     

On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

An extension of Jung’s theorem

Theorem. Let a set X?Rⁿ have unit circumradius and let B be the unit ball containing X. Put C =conv \(\bar X\) D =diam C (=diam X), k =dim C,d _i = √(2i + 2)/i. Then: (i) D∈[d_n, 2]; (ii) k≧m where m∈{2,3,...,n} satisfies D∈[d_m, d_m?1) (d_i decreases by i); (iii) In case k=m (by (ii), this is always the case when m=n), C contains a k-simplex Δ such that: (α) its vertices are on δB; (β) the centre of B belongs toint Δ; (γ) the inequalitiesλ _k (D) ≦l ≦D with $$\lambda _k (D) = D\sqrt {\frac{{4k - 2D^2 (k - 1)}}{{2 - (k - 2)(D^2 - 2)}}, D \in (d_k ,d_{k - 1} )} $$ are unimprovable estimates for length l of any edge of Δ.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(?₊), we introduce the modified strong dyadic integral J _α and the fractional derivative D ^(α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(?₊). We find a countable set of eigenfunctions of the operators D ^(α) and J _α, α > 0. We also prove the relations D ^(α)(J _α(f)) = f and J _α(D ^(α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$ . We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$ , where L _{J α} is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$ . Moreover, for a function f ∈ L(?₊) and a given point x ∈ ?₊, we introduce the modified dyadic derivative d ^(α)(f)(x) and the modified dyadic integral j _α(f)(x). We prove the relations d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.