On the location of the Weyl circles

1981 ◽  
Vol 88 (3-4) ◽  
pp. 345-356 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Riccati Equations ◽  
Sturm Liouville ◽  
Linear Integral ◽  
Linear Integral Equations

SynopsisThe paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.

scholarly journals Inverse problems for Sturm – Liouville operator. With potential functions from L2[0,π]

2020 ◽  
Vol 49 ◽  
pp. 28-38
Author(s):  
Biljana Vojvodić ◽  
◽  
Nataša Pavlović Komazec ◽  
Keyword(s):  
Inverse Problems ◽  
Boundary Value Problems ◽  
Spectral Problem ◽  
Integral Equations ◽  
Liouville Operator ◽  
Inverse Spectral Problem ◽  
Potential Functions ◽  
Sturm Liouville ◽  
Linear Integral ◽  
Linear Integral Equations

This paper deals with non-self-adjoint second-order differential operatorswith two constant delays. We consider four boundary value problems 𝐷𝑖,𝑘,𝑖=0,1,𝑘=1,2−𝑦′′(𝑥)+𝑞1(𝑥)𝑦(𝑥−𝜏1)+(−1)𝑖𝑞2(𝑥)𝑦(𝑥−𝜏2)=𝜆𝑦(𝑥),𝑥∈[0,𝜋]𝑦′(0)−ℎ𝑦(0)=0, 𝑦′(𝜋)+𝐻𝑘𝑦(𝜋)=0,where𝜋3≤𝜏2<𝜋2≤2𝜏2≤𝜏1<𝜋, ℎ,𝐻1,𝐻2∈ 𝑅\{0} and 𝜆 is a spectral parameter. We assumethat 𝑞1,𝑞2are real-valued potential functions from𝐿2[0,𝜋]such that 𝑞1(𝑥)=0,𝑥∈[0,𝜏1)and𝑞2(𝑥)=0,𝑥∈[0,𝜏2).The inverse spectral problem of recovering operators from their spectra hasbeen studied. We provethat delays 𝜏1,𝜏2and parameters ℎ,𝐻1,𝐻2are uniquely determined from thespectra. Then we prove that potentials are uniquely determined by Volterra linear integral equations.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

Integral equations and relations for Lamé functions and ellipsoidal wave functions

1968 ◽  
Vol 64 (1) ◽  
pp. 113-126 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Wave Function ◽  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Function Theory ◽  
Natural Generalization ◽  
Wave Functions ◽  
Linear Integral ◽  
First Time ◽  
Linear Integral Equations

AbstractNon-linear integral equations and relations, whose nuclei in all cases is the ‘potential’ Green's function, satisfied by Lamé polynomials and Lamé functions of the second kind are discussed. For these functions certain techniques of analysis are described and these find their natural generalization in ellipsoidal wave-function theory. Here similar integral equations are constructed for ellipsoidal wave functions of the first and third kinds, the nucleus in each case now being the ‘free space’ Green's function. The presence of ellipsoidal wave functions of the second kind is noted for the first time. Certain possible generalizations of the techniques and ideas involved in this paper are also discussed.

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