# Infinitely many singular interactions on noncompact manifolds

## Abstract

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004). - Highlights: • Schrödinger-operator for infinitely many singular interactions on noncompact manifolds. • Proof of the finiteness of the ground-state energy.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22451174

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 356; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; DELTA FUNCTION; GEOMETRY; GROUND STATES; KERNELS; MATHEMATICAL MANIFOLDS; RIEMANN SPACE; TWO-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Kaynak, Burak Tevfik, E-mail: burak.kaynak@boun.edu.tr, and Turgut, O. Teoman, E-mail: turgutte@boun.edu.tr.
```*Infinitely many singular interactions on noncompact manifolds*. United States: N. p., 2015.
Web. doi:10.1016/J.AOP.2015.03.016.

```
Kaynak, Burak Tevfik, E-mail: burak.kaynak@boun.edu.tr, & Turgut, O. Teoman, E-mail: turgutte@boun.edu.tr.
```*Infinitely many singular interactions on noncompact manifolds*. United States. doi:10.1016/J.AOP.2015.03.016.

```
Kaynak, Burak Tevfik, E-mail: burak.kaynak@boun.edu.tr, and Turgut, O. Teoman, E-mail: turgutte@boun.edu.tr. Fri .
"Infinitely many singular interactions on noncompact manifolds". United States. doi:10.1016/J.AOP.2015.03.016.
```

```
@article{osti_22451174,
```

title = {Infinitely many singular interactions on noncompact manifolds},

author = {Kaynak, Burak Tevfik, E-mail: burak.kaynak@boun.edu.tr and Turgut, O. Teoman, E-mail: turgutte@boun.edu.tr},

abstractNote = {We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004). - Highlights: • Schrödinger-operator for infinitely many singular interactions on noncompact manifolds. • Proof of the finiteness of the ground-state energy.},

doi = {10.1016/J.AOP.2015.03.016},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 356,

place = {United States},

year = {2015},

month = {5}

}