# Bounds on the density of states and the spectral gap in ${\mathrm{CFT}}_{2}$

## Abstract

We improve the recently discovered upper and lower bounds on the O(1) correction to the Cardy formula for the density of states integrated over an energy window (of width 2 δ), centered at high energy in two-dimensional unitary and modular invariant conformal field theory. We prove optimality of the lower bound for δ → 1^{–}. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than 1, demonstrating it to be 1. Furthermore, a systematic method is provided to establish a limit on how tight the bound on the O(1) correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of California, San Diego, CA (United States); Institute for Advanced Study, Princeton, NJ (United States)

- Sponsoring Org.:
- Simons Foundation; Ambrose Monell Foundation; USDOE Office of Science (SC)

- OSTI Identifier:
- 1630791

- Alternate Identifier(s):
- OSTI ID: 1801928

- Grant/Contract Number:
- SC0009919; SC0009988; 568420

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 101 Journal Issue: 10; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Astronomy & Astrophysics; Physics

### Citation Formats

```
Ganguly, Shouvik, and Pal, Sridip. Bounds on the density of states and the spectral gap in CFT 2. United States: N. p., 2020.
Web. doi:10.1103/PhysRevD.101.106022.
```

```
Ganguly, Shouvik, & Pal, Sridip. Bounds on the density of states and the spectral gap in CFT 2. United States. https://doi.org/10.1103/PhysRevD.101.106022
```

```
Ganguly, Shouvik, and Pal, Sridip. Tue .
"Bounds on the density of states and the spectral gap in CFT 2". United States. https://doi.org/10.1103/PhysRevD.101.106022.
```

```
@article{osti_1630791,
```

title = {Bounds on the density of states and the spectral gap in CFT 2},

author = {Ganguly, Shouvik and Pal, Sridip},

abstractNote = {We improve the recently discovered upper and lower bounds on the O(1) correction to the Cardy formula for the density of states integrated over an energy window (of width 2 δ), centered at high energy in two-dimensional unitary and modular invariant conformal field theory. We prove optimality of the lower bound for δ → 1–. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than 1, demonstrating it to be 1. Furthermore, a systematic method is provided to establish a limit on how tight the bound on the O(1) correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.},

doi = {10.1103/PhysRevD.101.106022},

journal = {Physical Review D},

number = 10,

volume = 101,

place = {United States},

year = {2020},

month = {5}

}

https://doi.org/10.1103/PhysRevD.101.106022

*Citation information provided by*

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