# ASYMPTOTIC SOLUTION OF EIGENVALUE PROBLEMS

## Abstract

A method is presented for the constrnction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential equations. It is an adaptation to bounded domains of the method previously devised to deduce the corrected Bohr-Sommerfeld quantum conditions. When applied to the reduced wave equation in various domains for which the exact solutions are known, it yields precisely the asymptotic forms of those solutions. It was applied to an arbitrary convex plane domain for which the exact solutions are not known. Two types of solutions were found, called the "whispering gallery" and "bouncing ball" modes. Applications were made to the Schroedinger equation. (auth)

- Authors:

- Publication Date:

- Research Org.:
- New York Univ.; Stevens Inst. of Tech., Hoboken, N.J.

- OSTI Identifier:
- 4172875

- NSA Number:
- NSA-14-007924

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York) (U.S.)

- Additional Journal Information:
- Journal Volume: Vol: 9; Other Information: Orig. Receipt Date: 31-DEC-60

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

- Subject:
- PHYSICS; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; EIGENFUNCTIONS; EIGENVALUES; EQUATIONS; MATHEMATICS; QUANTUM MECHANICS

### Citation Formats

```
Keller, J.B., and Rubinow, S.I.
```*ASYMPTOTIC SOLUTION OF EIGENVALUE PROBLEMS*. Country unknown/Code not available: N. p., 1960.
Web. doi:10.1016/0003-4916(60)90061-0.

```
Keller, J.B., & Rubinow, S.I.
```*ASYMPTOTIC SOLUTION OF EIGENVALUE PROBLEMS*. Country unknown/Code not available. doi:10.1016/0003-4916(60)90061-0.

```
Keller, J.B., and Rubinow, S.I. Fri .
"ASYMPTOTIC SOLUTION OF EIGENVALUE PROBLEMS". Country unknown/Code not available. doi:10.1016/0003-4916(60)90061-0.
```

```
@article{osti_4172875,
```

title = {ASYMPTOTIC SOLUTION OF EIGENVALUE PROBLEMS},

author = {Keller, J.B. and Rubinow, S.I.},

abstractNote = {A method is presented for the constrnction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential equations. It is an adaptation to bounded domains of the method previously devised to deduce the corrected Bohr-Sommerfeld quantum conditions. When applied to the reduced wave equation in various domains for which the exact solutions are known, it yields precisely the asymptotic forms of those solutions. It was applied to an arbitrary convex plane domain for which the exact solutions are not known. Two types of solutions were found, called the "whispering gallery" and "bouncing ball" modes. Applications were made to the Schroedinger equation. (auth)},

doi = {10.1016/0003-4916(60)90061-0},

journal = {Annals of Physics (New York) (U.S.)},

number = ,

volume = Vol: 9,

place = {Country unknown/Code not available},

year = {1960},

month = {1}

}

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