# Location of the essential spectrum in curved quantum layers

## Abstract

We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of the cross-section along the manifold is asymptotically parallel. For low dimensions and codimension, the result applies to the location of propagating states in nanostructures under physically natural conditions.

- Authors:

- Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Řež (Czech Republic)
- Department of Mathematics, University of California, Irvine, California 92697 (United States)

- Publication Date:

- OSTI Identifier:
- 22306028

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CROSS SECTIONS; EUCLIDEAN SPACE; LAPLACIAN; MATHEMATICAL MANIFOLDS; NANOSTRUCTURES; SPECTRA

### Citation Formats

```
Krejčiřík, David, E-mail: krejcirik@ujf.cas.cz, and Lu, Zhiqin, E-mail: zlu@uci.edu.
```*Location of the essential spectrum in curved quantum layers*. United States: N. p., 2014.
Web. doi:10.1063/1.4893035.

```
Krejčiřík, David, E-mail: krejcirik@ujf.cas.cz, & Lu, Zhiqin, E-mail: zlu@uci.edu.
```*Location of the essential spectrum in curved quantum layers*. United States. doi:10.1063/1.4893035.

```
Krejčiřík, David, E-mail: krejcirik@ujf.cas.cz, and Lu, Zhiqin, E-mail: zlu@uci.edu. Fri .
"Location of the essential spectrum in curved quantum layers". United States.
doi:10.1063/1.4893035.
```

```
@article{osti_22306028,
```

title = {Location of the essential spectrum in curved quantum layers},

author = {Krejčiřík, David, E-mail: krejcirik@ujf.cas.cz and Lu, Zhiqin, E-mail: zlu@uci.edu},

abstractNote = {We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of the cross-section along the manifold is asymptotically parallel. For low dimensions and codimension, the result applies to the location of propagating states in nanostructures under physically natural conditions.},

doi = {10.1063/1.4893035},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}

DOI: 10.1063/1.4893035

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