On the divergence of expansions in systems of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville operator with an arbitrary complex-valued potential q(x) ?? L ₁(0, ??) and with degenerate boundary conditions. We show that, under some additional condition, the system of root functions of that operator is not a basis in the space L ₂(0, ??).     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

On the completeness of the system of root functions of the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential q(x) ∈ L ₁(0, π) and with degenerate boundary conditions. We obtain sufficient conditions for the completeness of the system of eigenfunctions and associated functions of this operator.     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

Some consequences of the general Riemann hypothesis in the comparative theory of primes

Several effective results are proved about oscillatory properties of π(x, q, l₁) − π(x, q, l₂) and related functions assuming the General Riemann Hypothesis and the absence of real zeros of L-functions.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

On a two-point boundary value problem for the Sturm-Liouville operator with a nonclassical asymptotics of the spectrum

We consider a spectral problem generated by a Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We prove the existence of potentials q(x) ∈ L ₂(0, π) such that the multiplicities of the eigenvalues λ _n monotonically tend to infinity as n → ∞.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

A uniform finite element method for a conservative singularly perturbed problem

We examine the problem ϵ(p(x)u′) + (q(x)u)′ − r(x)u = f(x) for 0 < x < 1, p>0, q>0, r ⩾ 0; p, q, r and f in C²[0, 1], ϵ in (0, 1], u(0) and u(1) given. Existence of a unique solution u and bounds on u and its derivatives are obtained. Using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem (i.e., the nodal errors are bounded by Ch², where C is independent of h and ϵ). With a natural choice of trial functions, uniform first order accuracy is obtained in the L^∞(0, 1) norm. Using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the L¹(0, 1) norm.     

Basis Properties of the System of Root Functions of the Sturm–Liouville Operator with Degenerate Boundary Conditions: I

The spectral problem for the Sturm–Liouville operator with an arbitrary complex-valued potential q(x) of the class L¹(0, π) and degenerate boundary conditions is considered. We prove that the system of root functions of this operator is not a basis in the space L²(0, π).     

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L ₂(?1, 1) by the operation with involution of the form Lu = ?u″(x)+αu″(?x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (?1, 1), ν(x) is an absolutely continuous involution of the segment [?1, 1] and the coefficients q(x) and qν(x) are summable functions on (?1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L ₂(?1, 1) for the system of the root functions of the operator.     

Schrödinger operators with singular potentials

Recently B. Simon proved a remarkable theorem to the effect that the Schrödinger operatorT=?Δ+q(x) is essentially selfadjoint onC ₀ ^∞ (R ^m if 0≦q ∈L ²(R ^m). Here we extend the theorem to a more general case,T=?Σ _j ^=1/m (?/?x _j ?ib _j(x))² +q ₁(x) +q ₂(x), whereb _j, q₁,q ₂ are real-valued,b _j ∈C(R ^m),q ₁ ∈L _loc ² (R ^m),q ₁(x)≧?q*(|x|) withq*(r) monotone nondecreasing inr ando(r ²) asr → ∞, andq ₂ satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq ₁ is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.     

Equiconvergence of the trigonometric Fourier series and the expansion in eigenfunctions of the Sturm-Liouville operator with a distribution potential

We consider the Sturm-Liouville operator L = ?d ²/dx ² + q(x) with the Dirichlet boundary conditions in the space L ₂[0, π] under the assumption that the potential q(x) belongs to W ₂ ^?1 [0, π]. We study the problem of uniform equiconvergence on the interval [0, π] of the expansion of a function f(x) in the system of eigenfunctions and associated functions of the operator L and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function f(x) of class L ₁. We also consider the case of potentials belonging to the scale of Sobolev spaces W ₂ ^?θ [0, π] with ½ < θ ≤ 1. We show that if the antiderivative u(x) of the potential belongs to some space W ₂ ^θ [0, π] with 0 < θ < 1/2, then, for any function in the space L ₂[0, π], the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing u(x). We also give an explicit estimate for the rate of equiconvergence.     

On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u₀(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a₀>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a₀>0 in Ω, and A is bounded in if a(x)?0 in Ω.     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).