Layer potentials for Schrödinger operators and pseudodifferential operators on manifolds with cylindrical ends

Abstract

The method of layer potentials is one of the oldest methods to solve boundary value problems. While it is slightly more complicated to use than the energy (or variational) method, it has some additional advantages. In this paper, we study the basic (single and double) layer potentials for the Schrödinger operator ∆_V := ∆ -V , where ∆ := -∇ * ∇ is the Laplace-Beltrami operator and V is a suitable scalar potential on a domain with straight cylindrical ends Ω (sometimes called a "waveguide"). First, under suitable positivity assumption at infinity on V , we prove that ∆ V is Fredholm on L^2 (M ). This allows us then to define the single-and double-layer potential operators S and K, which we prove are suitable pseudodifferential operators. Under suitable further positivity assumptions on V (depending on the operator studied), we prove that S, 1/2 + K, and -1/2 + K * are invertible. The proofs of these results require us to recall and develop many other related tools. In particular, we recall an "algebra tool kit" to deal with limit and jump relations of pseudodifferential layer potential operators on manifolds with straight cylindrical ends. We use these results to devise a new approach to the limit and jump relations of the layer potential operators associated to our Schrödinger operator ∆ V . The proofs are based on the fact that S and K are in Ψ^{-1}_{ess} (∂Ω) (i.e. they are order -1 pseudodifferential operators essentially translation invariant at infinity). As applications, we prove the well-posedness of the Dirichlet and Neumann problems, including a Neumann problem for a semi-linear Schrödinger equation with small data on a domain with cylindrical ends.

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BibTeX RIS

BibTeX

@misc{nistor2026layer,
  title = {{Layer potentials for Schrödinger operators and pseudodifferential operators on manifolds with cylindrical ends}},
  author = {Victor Nistor and Mirela Kohr and Wolfgang L Wendland},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05774326v1},
  note = {CC BY-NC 4.0},
}

RIS

TY  - UNPB
TI  - Layer potentials for Schrödinger operators and pseudodifferential operators on manifolds with cylindrical ends
AU  - Victor Nistor
AU  - Mirela Kohr
AU  - Wolfgang L Wendland
PY  - 2026
DA  - 2026/10/01
PB  - HAL
UR  - https://hal.science/hal-05774326v1
AB  - The method of layer potentials is one of the oldest methods to solve boundary value problems. While it is slightly more complicated to use than the energy (or variational) method, it has some additional advantages. In this paper, we study the basic (single and double) layer potentials for the Schrödinger operator ∆_V := ∆ -V , where ∆ := -∇ * ∇ is the Laplace-Beltrami operator and V is a suitable scalar potential on a domain with straight cylindrical ends Ω (sometimes called a "waveguide"). First, under suitable positivity assumption at infinity on V , we prove that ∆ V is Fredholm on L^2 (M ). This allows us then to define the single-and double-layer potential operators S and K, which we prove are suitable pseudodifferential operators. Under suitable further positivity assumptions on V (depending on the operator studied), we prove that S, 1/2 + K, and -1/2 + K * are invertible. The proofs of these results require us to recall and develop many other related tools. In particular, we recall an "algebra tool kit" to deal with limit and jump relations of pseudodifferential layer potential operators on manifolds with straight cylindrical ends. We use these results to devise a new approach to the limit and jump relations of the layer potential operators associated to our Schrödinger operator ∆ V . The proofs are based on the fact that S and K are in Ψ^{-1}_{ess} (∂Ω) (i.e. they are order -1 pseudodifferential operators essentially translation invariant at infinity). As applications, we prove the well-posedness of the Dirichlet and Neumann problems, including a Neumann problem for a semi-linear Schrödinger equation with small data on a domain with cylindrical ends.
ER  -

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