Trace formulas for Schrödinger operators on star graphs with general matching conditions

2018 ◽  
Vol 51 (36) ◽  
pp. 365301
Author(s):  
Muhammad Usman ◽  
Ali Ashher Zaidi
Keyword(s):  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Trace Formulas ◽  
Matching Conditions ◽  
Star Graphs

scholarly journals On inverse topology problem for Laplace operators on graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 230-236
Author(s):  
Yu.Yu. Ershova ◽  
I.I. Karpenko ◽  
A.V. Kiselev
Keyword(s):  
Additional Assumption ◽  
Necessary Conditions ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Metric Graphs ◽  
Laplace Operators ◽  
Matching Conditions ◽  
Inverse Topology ◽  
Delta Type ◽  
Laplacian Operators

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ type. Under one additional assumption, the inverse topology problem is treated. Using the apparatus of boundary triples, we generalize and extend existing results on necessary conditions of isospectrality of two Laplacians defined on different graphs. A result is also given covering the case of Schrodinger operators.

TRACE FORMULAS AND CONSERVATION LAWS FOR NONLINEAR EVOLUTION EQUATIONS

1994 ◽  
Vol 06 (01) ◽  
pp. 51-95 ◽  
Author(s):  
F. GESZTESY ◽  
H. HOLDEN
Keyword(s):  
Conservation Laws ◽  
Long Range ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Schrödinger Operators ◽  
Linear Operators ◽  
Nonlinear Evolution Equations ◽  
Schrodinger Operators ◽  
Trace Formulas ◽  
One Dimensional

New trace formulas for linear operators associated with Lax pairs or zero-curvature representations of completely integrable nonlinear evolution equations and their relation to (polynomial) conservation laws are established. We particularly study the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the sine–Gordon equation, and the infinite Toda lattice though our methods apply to any element of the AKNS–ZS class. In the KdV context, we especially extend the range of validity of the infinite sequence of conservation laws to certain long-range situations in which the underlying one-dimensional Schrödinger operator has infinitely many (negative) eigenvalues accumulating at zero. We also generalize inequalities on moments of the eigenvalues of Schrödinger operators to this long-range setting. Moreover, our contour integration approach naturally leads to higher-order Levinson-type theorems for Schrödinger operators on the line.

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