scholarly journals On Boundary Conditions for Sturm-Liouville Differential Operators in the Direct Sum Spaces

1999 ◽  
Vol 29 (3) ◽  
pp. 873-892
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Direct Sum ◽  
Sturm Liouville

scholarly journals On the product of self-adjoint Sturm-Liouville differential operators in direct sum spaces

2006 ◽  
Vol 37 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Direct Sum ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1,\tau_2, \ldots, \tau_n $, with real coefficients on any finite number of intervals are studied in the setting of the direct sum of the $ L_w^2 $-spaces of functions defined on each of the separate intervals. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, it is an exact parallel of that in the regular case. This characterization is an extension of those obtained in [6], [7], [8], [9], [12], [14] and [15].

scholarly journals Characterization of Self-Adjoint Domains for Two-Interval Even Order Singular C-Symmetric Differential Operators in Direct Sum Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qinglan Bao ◽  
Xiaoling Hao ◽  
Jiong Sun
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Direct Sum ◽  
Even Order ◽  
Special Case ◽  
Symmetric Differential Operators

This paper is concerned with the characterization of all self-adjoint domains associated with two-interval even order singular C-symmetric differential operators in terms of boundary conditions. The previously known characterizations of Lagrange symmetric differential operators are a special case of this one.

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

scholarly journals TWO-INTERVAL EVEN-ORDER DIFFERENTIAL OPERATORS IN MODIFIED HILBERT SPACES

2011 ◽  
Vol 85 (2) ◽  
pp. 241-260
Author(s):  
JIANQING SUO ◽  
WANYI WANG
Keyword(s):  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Order Equation ◽  
Inner Product ◽  
Coupling Matrix ◽  
Direct Sum ◽  
Even Order

AbstractBy modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which an even-order equation is defined, we generate self-adjoint realisations for boundary conditions with any real coupling matrix which are much more general than the coupling matrices from the ‘unmodified’ theory.

scholarly journals Inverse problems for differential operators with nonseparated boundary conditions in the central symmetric case

2017 ◽  
Vol 48 (4) ◽  
pp. 377-387 ◽  
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Symmetric Case ◽  
Nonseparated Boundary Conditions ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Liouville Operators

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.

scholarly journals On the boundary conditions for products of Sturm-Liouville differential operators

2001 ◽  
Vol 32 (3) ◽  
pp. 187-199
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1, \tau_2, \ldots, \tau_n $ with real coefficients are considered on the interval $ I = (a,b) $, $ - \infty \le a < b \le \infty $. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximan domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained in [6], [8], [11-12], [14] and [15].

scholarly journals On the domain of selfadjoint extension of the product of Sturm-Liouville differential operators

2003 ◽  
Vol 2003 (11) ◽  
pp. 695-709
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Selfadjoint Extension ◽  
Sesquilinear Form ◽  
Sturm Liouville

The second-order symmetric Sturm-Liouville differential expressionsτ1,τ2,…,τnwith real coefficients are considered on the intervalI=(a,b),−∞≤a<b≤∞. It is shown that the characterization of singular selfadjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, and it is an exact parallel of the regular case. This characterization is an extension of those obtained by Everitt and Zettl (1977), Hinton, Krall, and Shaw (1987), Ibrahim (1999), Krall and Zettl (1988), Lee (1975/1976), and Naimark (1968).

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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