Estimations of Solutions of the Sturm– Liouville Equation with Respect to a Spectral Parameter

This paper is concerned with estimations of solutions of the Sturm–Liouville equation $$\big(p(x)y'(x)\big)'+\Big(\mu^2 -2i\mu d(x)-q(x)\Big)\rho(x)y(x)=0, \ \ x\in[0,1],$$ ( p ( x ) y ' ( x ) ) ' + ( μ 2 - 2 i μ d ( x ) - q ( x ) ) ρ ( x ) y ( x ) = 0 , x ∈ [ 0 , 1 ] , where ${\mu\in\mathbb{C}}$ μ ∈ C is a spectral parameter. We assume that the strictly positive function ${\rho\in L_{\infty}[0,1]}$ ρ ∈ L ∞ [ 0 , 1 ] is of bounded variation, ${p\in W^1_1[0,1]}$ p ∈ W 1 1 [ 0 , 1 ] is also strictly positive, while ${d\in L_1[0,1]}$ d ∈ L 1 [ 0 , 1 ] and ${q\in L_1[0,1]}$ q ∈ L 1 [ 0 , 1 ] are real functions. The main result states that for any r > 0 there exists a constant c _r such that for any solution y of the Sturm–Liouville equation with μ satisfying ${|{\rm Im}\, \mu|\leq r}$ | Im μ | ≤ r , the inequality ${\|y(\cdot,\mu)\|_C\leq c_r\|y(\cdot,\mu)\|_{L_1}}$ ∥ y ( · , μ ) ∥ C ≤ c r ∥ y ( · , μ ) ∥ L 1 is true. We apply our results to a problem of vibrations of an inhomogeneous string of length one with damping, modulus of elasticity and potential, rewritten in an operator form. As a consequence, we obtain that the operator acting on a certain energy Hilbert space is the generator of an exponentially stable C ₀-semigroup.     

On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Arithmetically Self-Similar Weight of a Generalized Cantor Type

Spectral asymptotics of the Sturm–Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method for estimating the eigenvalue counting function. This makes it possible to consider a much wider class of self-similar measures.     

Inverse problems for the matrix Sturm–Liouville equation with a Bessel-type singularity

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems.     

Equivalent Sets of Solutions of the Klein–Gordon Equation with a Constant Electric Field

We argue extensively in favor of our earlier choice of the in and out states (among the solutions of a wave equation with one-dimensional potential). In this connection, we study the nonstationary and stationary families of complete sets of solutions of the Klein–Gordon equation with a constant electric field. A nonstationary set _Pv consists of the solutions with the quantum number p _v=p ⁰ v–p₃. It can be obtained from the nonstationary set _P3 with the quantum number p ₃ by a boost along the x ₃ axis (in the direction of the electric field) with the velocity –v. By changing the gauge, we can bring the solutions in all sets to the same potential without changing quantum numbers. Then the transformations of solutions in one set (with the quantum number p _v) to the solutions in another set (with the quantum number p _v) have group properties. The stationary solutions and sets have the same properties as the nonstationary ones and are obtainable from stationary solutions with the quantum number p ⁰ by the same boost. It turns out that each set can be obtained from any other by gauge manipulations. All sets are therefore equivalent, and the classification (i.e., assigning the frequency sign and the in and out indices) in any set is determined by the classification in the set _P3, where it is obvious.     

Infinitely Many Solutions for Systems of Sturm–Liouville Boundary Value Problems

In this paper, the authors obtain the existence of infinitely many classical solutions to the boundary value system with Sturm–Liouville boundary conditions $$\left\{\begin{array}{ll}-(\phi_{p_i}(u_{i}^\prime))^\prime = \lambda F_{u_{i}}(x,u_{1},\ldots,u_{n})h_{i}(u^\prime_i)\quad {\rm in} \, (a,b), \\ \alpha_iu_{i}(a)-\beta_iu^ \prime_{i}(a)=0, \quad \gamma_iu_{i}(b)+\sigma_iu^\prime_{i}(b)=0, \end{array}\quad{i = 1, \ldots , n.} \right.$$ Critical point theory and Ricceri’s variational principle are used in the proofs.     

On the determination of the impulsive Sturm–Liouville operator with the eigenparameter-dependent boundary conditions

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h₁,h₂, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H₁,H₂, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.     

On the solvability of inverse Sturm–Liouville problems with self-adjoint boundary conditions

Theorems on the existence and uniqueness of a solution of the inverse Sturm–Liouville problem with self-adjoint nonseparated boundary conditions are proved. As spectral data two spectra and two eigenvalues are used. The theorems generalize the Levitan–Gasymov solvability theorem and Borg’s uniqueness theorem to the case of general boundary conditions.     

On the uniqueness of the solution of dual equation of a singular Sturm–Liouville problem

In this paper we consider differential systems having a singularity and one turning point. First, by a replacement, we transform the system to a linear second-order equation of Sturm–Liouville type with a singularity. Using the infinite product representation of solutions provided in [8], we obtain the dual equation, then we investigate the uniqueness of the solution for the dual equation of the inverse spectral problem of Sturm–Liouville equation. This result is necessary for expressing inverse problem of indefinite Sturm–Liouville equation.     

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

Logarithmic Asymptotics of the Nonlinear Cauchy–Riemann–Beltrami Equation

Ukrainian Mathematical Journal - We study regular solutions of the nonlinear Cauchy–Riemann–Beltrami equation for the logarithmic asymptotics in terms of the lower limits and solve an...     

Spectral Properties of Discontinuous Sturm–Liouville Problems with a Finite Number of Transmission Conditions

In this paper we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of the considered problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenelements of A are complete in H.     

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.     

On the Estimates of Periodic Eigenvalues of Sturm–Liouville Operators with Trigonometric Polynomial Potentials

Mathematical Notes - We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with periodic and antiperiodic boundary conditions for special potentials that are...     

Asymptotic Behavior of Bifurcation Branch of Positive Solutions for Semilinear Sturm–Liouville Problems

We consider the nonlinear eigenvalue problem

Multiple fixed-sign solutions for a system of difference equations with Sturm–Liouville conditions

We consider the following system of difference equations:${\mathrm{\Delta }}^m{u}_i(k)+P_i(k,u_1(k),u_2(k),\dots ,u_n(k))=0,k\in \{0,1,\dots ,N\},i=1,2,\dots ,n$together with Sturm–Liouville boundary conditions${\mathrm{\Delta }}^j{u}_i(0)=0,0⩽j⩽m-3\text{,}$$\zeta {\mathrm{\Delta }}^{m-2}{u}_i(0)-\eta {\mathrm{\Delta }}^{m-1}{u}_i(0)=0,\omega {\mathrm{\Delta }}^{m-2}{u}_i(N+1)+\delta {\mathrm{\Delta }}^{m-1}{u}_i(N+1)=0\text{,}$where $m⩾2,N⩾m-1,\zeta >0,\omega >0,\eta ⩾0,\delta ⩾\omega ,\zeta \omega (N+1)+\zeta \delta +\eta \omega >0$. By using two different fixed point theorems, we develop criteria for the existence of three solutions of the system which are of fixed signs on $\{0,1,\dots ,N+m\}$. Examples are also included to illustrate the results obtained.     

Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.