# A complex guided spectral transform Lanczos method for studying quantum resonance states

## Abstract

A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths and wavefunctions. The algorithm comprises of two layers of complex-symmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos (cFOL) polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigen-pairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a root-searching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO₂, and compared to previous calculations.

- Authors:

- Brookhaven National Lab. (BNL), Upton, NY (United States). Dept. of Chemistry

- Publication Date:

- Research Org.:
- Brookhaven National Laboratory (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1182481

- Alternate Identifier(s):
- OSTI ID: 1226656

- Report Number(s):
- BNL-107310-2014-JA

Journal ID: ISSN 0021-9606; JCPSA6; R&D Project: 04540; 04548; TRN: US1500504

- Grant/Contract Number:
- SC00112704

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 141; Journal Issue: 24; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; iterative diagonalization; complex symmetric matrix; spectral transform Lanczos

### Citation Formats

```
Yu, Hua-Gen.
```*A complex guided spectral transform Lanczos method for studying quantum resonance states*. United States: N. p., 2014.
Web. doi:10.1063/1.4905083.

```
Yu, Hua-Gen.
```*A complex guided spectral transform Lanczos method for studying quantum resonance states*. United States. doi:10.1063/1.4905083.

```
Yu, Hua-Gen. Sun .
"A complex guided spectral transform Lanczos method for studying quantum resonance states". United States.
doi:10.1063/1.4905083. https://www.osti.gov/servlets/purl/1182481.
```

```
@article{osti_1182481,
```

title = {A complex guided spectral transform Lanczos method for studying quantum resonance states},

author = {Yu, Hua-Gen},

abstractNote = {A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths and wavefunctions. The algorithm comprises of two layers of complex-symmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos (cFOL) polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigen-pairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a root-searching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO₂, and compared to previous calculations.},

doi = {10.1063/1.4905083},

journal = {Journal of Chemical Physics},

number = 24,

volume = 141,

place = {United States},

year = {Sun Dec 28 00:00:00 EST 2014},

month = {Sun Dec 28 00:00:00 EST 2014}

}

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