## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Schrödinger operators with $n$ positive eigenvalues: an explicit construction involving complex-valued potentials

#### Abstract

An explicit construction is provided for embedding $n$ positive eigenvalues in the spectrum of a Schrödinger operator on the half-line with a Dirichlet boundary condition at the origin. The resulting potential is of von Neumann-Wigner type, but can be real-valued as well as complex-valued.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 1 (2016), 7-12.

Dates
First available in Project Euclid: 28 December 2015

https://projecteuclid.org/euclid.pja/1451330560

Digital Object Identifier
doi:10.3792/pjaa.92.7

Mathematical Reviews number (MathSciNet)
MR3447743

Zentralblatt MATH identifier
0476.03047

#### Citation

Richard, Serge; Uchiyama, Jun; Umeda, Tomio. Schrödinger operators with $n$ positive eigenvalues: an explicit construction involving complex-valued potentials. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 1, 7--12. doi:10.3792/pjaa.92.7. https://projecteuclid.org/euclid.pja/1451330560

#### References

• S. Agmon, I. Herbst and S. Maad Sasane, Persistence of embedded eigenvalues, J. Funct. Anal. 261 (2011), no. 2, 451–477.
• W. O. Amrein, J. M. Jauch and K. B. Sinha, Scattering theory in quantum mechanics, W. A. Benjamin, Inc., Reading, MA, 1977.
• M. Arai and J. Uchiyama, On the von Neumann and Wigner potentials, J. Differential Equations 157 (1999), no. 2, 348–372.
• M. Ben-Artzi and A. Devinatz, Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys. 20 (1979), no. 4, 594–607.
• J. Cruz-Sampedro, I. Herbst and R. Martínez-Avendaño, Perturbations of the Wigner-von Neumann potential leaving the embedded eigenvalue fixed, Ann. Henri Poincaré 3 (2002), no. 2, 331–345.
• M. S. P. Eastham and H. Kalf, Schrödinger-type operators with continuous spectra, Research Notes in Mathematics, 65, Pitman, Boston, MA, 1982.
• I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304.
• M. Lukic, Schrödinger operators with slowly decaying Wigner-von Neumann type potentials, J. Spectr. Theory 3 (2013), no. 2, 147–169.
• H. E. Moses and S. F. Tuan, Potentials with zero scattering phase, Nuovo Cimento 13 (1959), no. 1, 197–206.
• M. Reed and B. Simon, Methods of modern mathematical physics. III: Scattering theory, Academic Press, New York, 1979.
• B. Simon, Some Schrödinger operators with dense point spectrum, Proc. Amer. Math. Soc. 125 (1997), no. 1, 203–208.
• J. Uchiyama, Simple construction of the Schrödinger operator having many positive eigenvalues, in Proceedings of the Fourth Workshop on Differential Equations, Chonnam National University, (Kwangju, 1999), 197–199.
• J. von Neumann and E. Wigner, Uber merkwürdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465–467.