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Cospectral Trees Indistinguishable by Scattering

Pistol, Mats Erik LU and Pivovarchik, Vyacheslav (2025) In Complex Analysis and Operator Theory 19(5).
Abstract

Let v1 and v2 be two distinct vertices of an equilateral tree T0. Let ϕN(i) (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T0 rooted at vi with Neumann conditions at the root and let ϕD(i) (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T0 with Dirichlet conditions at the root. We prove that if attaching any tree to T0 at the vertices v1 and v2 leads to cospectral trees and d(v1)=d(v2) then ϕN(λ)(1)≡ϕN(λ)(2) and ϕD(λ)(1)≡ϕD(λ)(1) (which means that the scattering is the same at v1 and v2).

Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Dirichlet boundary condition, Eigenvalue, Equilateral tree, Neumann boundary condition, Root, spectrum, Sturm-Liouville equation
in
Complex Analysis and Operator Theory
volume
19
issue
5
article number
104
publisher
Birkhäuser
external identifiers
  • scopus:105007919667
ISSN
1661-8254
DOI
10.1007/s11785-025-01730-6
language
English
LU publication?
yes
additional info
Publisher Copyright: © The Author(s) 2025.
id
1d012336-b9a8-40b9-ba23-b95f0a900795
date added to LUP
2025-12-11 15:27:18
date last changed
2025-12-15 13:27:41
@article{1d012336-b9a8-40b9-ba23-b95f0a900795,
  abstract     = {{<p>Let v1 and v2 be two distinct vertices of an equilateral tree T0. Let ϕN(i) (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T0 rooted at vi with Neumann conditions at the root and let ϕD(i) (i=1,2) be the characteristic functions of the Sturm-Liouville problem on T0 with Dirichlet conditions at the root. We prove that if attaching any tree to T0 at the vertices v1 and v2 leads to cospectral trees and d(v1)=d(v2) then ϕN(λ)(1)≡ϕN(λ)(2) and ϕD(λ)(1)≡ϕD(λ)(1) (which means that the scattering is the same at v1 and v2).</p>}},
  author       = {{Pistol, Mats Erik and Pivovarchik, Vyacheslav}},
  issn         = {{1661-8254}},
  keywords     = {{Dirichlet boundary condition; Eigenvalue; Equilateral tree; Neumann boundary condition; Root; spectrum; Sturm-Liouville equation}},
  language     = {{eng}},
  number       = {{5}},
  publisher    = {{Birkhäuser}},
  series       = {{Complex Analysis and Operator Theory}},
  title        = {{Cospectral Trees Indistinguishable by Scattering}},
  url          = {{http://dx.doi.org/10.1007/s11785-025-01730-6}},
  doi          = {{10.1007/s11785-025-01730-6}},
  volume       = {{19}},
  year         = {{2025}},
}