WKB expansion for systems of Schrödinger operators with crossing eigenvalues
(1997) In Asymptotic Analysis 14(1). p.1-48- Abstract
Let M be a compact, connected C∞ manifold with a C∞ Riemannian metric or let M=Rn with Euclidean metric. Let g denote the metric. On M consider the 2×2 system of Schrödinger operators
[TeX:] \[P=-h^{2}\Delta +V+h^{2}\mathcal{W},\qquadV=\left(\begin{array}[cc]v_{1}&0\\0&v_{2}\end{array}\right)\]where v1,v2∈C∞(M) are non-negative, and W is a C∞, first order formally selfadjoint differential operator with real coefficients. We study the eigenfunctions of P corresponding to the lowest eigenvalues in the semi-classical limit h→0. They are concentrated near a minimal geodesic γ with... (More)Let M be a compact, connected C∞ manifold with a C∞ Riemannian metric or let M=Rn with Euclidean metric. Let g denote the metric. On M consider the 2×2 system of Schrödinger operators
[TeX:] \[P=-h^{2}\Delta +V+h^{2}\mathcal{W},\qquadV=\left(\begin{array}[cc]v_{1}&0\\0&v_{2}\end{array}\right)\]where v1,v2∈C∞(M) are non-negative, and W is a C∞, first order formally selfadjoint differential operator with real coefficients. We study the eigenfunctions of P corresponding to the lowest eigenvalues in the semi-classical limit h→0. They are concentrated near a minimal geodesic γ with respect to the Agmon metric vg, v = min(v1, v2), connecting two non-degenerate zeros of v (wells). This metric is Lipschitz continuous but not C2 at [TeX:] $\varGamma =\{x\in M;\ v_{1}(x)=v_{2}(x)\}$. When the derivative of v1 - v2 along γ does not vanish on [TeX:] $\gamma \cap \varGamma $ and the intersection is transversal we obtain WKB expansions of the eigenfunctions. At [TeX:] $\gamma \cap \varGamma $ they are expressed in terms of derivatives Yk,ε=∂kYε/∂εk of suitable parabolic cylinder functions Yε. As an application of the WKB constructions we compute the splitting due to tunnelling of the lowest eigenvalues of P under a strong symmetry condition. (Less)
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- author
- Pettersson, Pelle LU
- organization
- publishing date
- 1997
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Asymptotic Analysis
- volume
- 14
- issue
- 1
- pages
- 1 - 48
- publisher
- I O S Press
- external identifiers
-
- scopus:1642357501
- ISSN
- 0921-7134
- DOI
- 10.3233/ASY-1997-14101
- language
- English
- LU publication?
- yes
- id
- a4c6ca4e-5758-41ce-bf11-06000c0a0d1f
- date added to LUP
- 2022-04-07 12:56:50
- date last changed
- 2023-05-02 04:02:27
@article{a4c6ca4e-5758-41ce-bf11-06000c0a0d1f,
abstract = {{<p class="first" style="box-sizing: border-box; margin: 0em 0px; color: rgb(65, 65, 65); font-family: "PT Serif", serif; font-size: 16px;">Let M be a compact, connected C∞ manifold with a C∞ Riemannian metric or let M=Rn with Euclidean metric. Let g denote the metric. On M consider the 2×2 system of Schrödinger operators</p>[TeX:] \[P=-h^{2}\Delta +V+h^{2}\mathcal{W},\qquadV=\left(\begin{array}[cc]v_{1}&0\\0&v_{2}\end{array}\right)\]where v1,v2∈C∞(M) are non-negative, and W is a C∞, first order formally selfadjoint differential operator with real coefficients. We study the eigenfunctions of P corresponding to the lowest eigenvalues in the semi-classical limit h→0. They are concentrated near a minimal geodesic γ with respect to the Agmon metric vg, v = min(v1, v2), connecting two non-degenerate zeros of v (wells). This metric is Lipschitz continuous but not C2 at [TeX:] $\varGamma =\{x\in M;\ v_{1}(x)=v_{2}(x)\}$. When the derivative of v1 - v2 along γ does not vanish on [TeX:] $\gamma \cap \varGamma $ and the intersection is transversal we obtain WKB expansions of the eigenfunctions. At [TeX:] $\gamma \cap \varGamma $ they are expressed in terms of derivatives Yk,ε=∂kYε/∂εk of suitable parabolic cylinder functions Yε. As an application of the WKB constructions we compute the splitting due to tunnelling of the lowest eigenvalues of P under a strong symmetry condition.}},
author = {{Pettersson, Pelle}},
issn = {{0921-7134}},
language = {{eng}},
number = {{1}},
pages = {{1--48}},
publisher = {{I O S Press}},
series = {{Asymptotic Analysis}},
title = {{WKB expansion for systems of Schrödinger operators with crossing eigenvalues}},
url = {{http://dx.doi.org/10.3233/ASY-1997-14101}},
doi = {{10.3233/ASY-1997-14101}},
volume = {{14}},
year = {{1997}},
}