On the resolvent of singular Sturm‐Liouville operators with transmission conditions

Abstract

In this article, we investigate the resolvent operator of singular Sturm‐Liouville problem with transmission conditions. We obtain integral representations for the resolvent of this operator in terms of the spectral function. Later, we discuss some properties of the resolvent operator, such as Hilbert‐Schmidt kernel property, compactness. Finally, we give a formula in terms of the spectral function for the Weyl‐Titchmarsh function of this problem.

On the resolvent of singular q-Sturm-Liouville operators

“…Historically, in 1910, H. Weyl was …rst obtained a representation theorem for the resolvent of Sturm-Liouville problem de…ned by (py 0 ) 0 + qy = y; x 2 (0; 1); where p; q are real-valued and p 1 ; q 2 L 1 loc [0; 1). Similar representation theorems were proved in [25,20,2,5,6,7].…”

“…where f 2 L 2 q [0; q n ]: Now, we shall show that the equality (7) satis…es the equation L(y) y(x) = f (x); x 2 (0; q n ) ( 2 C; Im 6 = 0) and the boundary conditions (4)- (5). From (7), we get y (x; ) = q q n (x; )…”

Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

“…Let λ m,q −n (where m, n ∈ N) denote the eigenvalues of the regular problem given by (3)-(5), and ϕ(x, λ) be the solution of the equation (3) satisfying the initial conditions in (6). The function ϕ m,q −n (x) = ϕ(x, λ m,q −n ) will be an eigenfunction corresponding to the eigenvalue λ m,q −n .…”

“…Usually, if we want to solve a partial differential equation using the Fourier method (i.e., the separation of variables) then we consider the problem of expanding an arbitrary function as a series of eigenfunctions. Hence the eigenfunction expanding problem has been studied extensively in the literature (see [2], [3], [4], [5], [6], [7], [13], [15], [16], [21], [22], [29], [30], [31], [34], [35], [36], [38], [39]).…”

The Transmittal-Characteristic Function of Three-Interval Periodic Sturm-Liouville Problem with Transmission Conditions

“…Boundary value problems including transmission conditions appears in many fields of natural sciences. Recently, such type of transmission problems have been an important topic in theoretical and applied mathematics (see, [17][18][19][20][21][22][23][24][25][26][27][28]). In this study we will investigate some basic spectral properties of a new type periodic Sturm-Liouville problems.…”