Acta Polytechnica


Acta Polytechnica 53(3):306–307, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

SPECTRAL SINGULARITIES DO NOT CORRESPOND
TO BOUND STATES IN THE CONTINUUM

Ali Mostafazadeh∗

Department of Mathematics, Koç University, 34450 Sarıyer, Istanbul, Turkey
∗ corresponding author: amostafazadeh@ku.edu.tr

Abstract. We show that, contrary to a claim made in arXiv:1011.0645, the von Neumann-Winger
bound states that lie in the continuum of the scattering states are fundamentally different from
Naimark’s spectral singularities.

Keywords: spectral singularity, bound state in continuum, resonance.

In 1927, von Neumann and Wigner constructed a
spherically symmetric scattering potential that sup-
ported a bound state with a positive energy [1]. This
corresponded to a genuine, square integrable solu-
tion of the time-independent Schrödinger equation.
Because the continuous spectrum of the Schrödinger
operator coincided with the set of nonnegative real
numbers, this bound state was called “a bound state
in the continuum.” This class of bound states was
subsequently studied by various authors, notably Still-
inger and Herrick [2], Gazdy [3], and Friedrich and
Wintgen [4] who interpreted them as resonances with
a zero width.
In 2009, the present author revealed the physical

meaning and possible applications of the mathemat-
ical concept of a spectral singularity [5]. This is a
generic feature of complex scattering potentials that
was discovered by Naimark in the 1950s [6] and stud-
ied thoroughly by mathematicians for more than half a
century [7]. Because spectral singularities correspond
to poles of the reflection and transmission coefficients
(and of the S-matrix) and belong to the continuous
spectrum of the Schrödinger operator, which is real
and nonnegative, they can also be interpreted as zero-
width resonances [5]. This has led to the claim [8]
that states associated with spectral singularities are
nothing but the bound states in the continuum.

A quick look at the properties of spectral singular-
ities show, that they correspond to scattering solu-
tions of the time-independent Schrödinger equation.
In particular, they are not square-integrable. This
is in contrast with bound states in the continuum,
which are associated with square-integrable solutions
of the time-independent Schrödinger equation. The
following are other major differences between spectral
singularities and bound states in the continuum.
(1.) A real scattering potential can never support a
spectral singularity [7]. This is certainly not the
case for bound states in the continuum. In fact, the
scattering potentials introduced by von Neumann
and Wigner [1] and others [2, 3] that admit a bound
state in the continuum are real.

(2.) Spectral singularities appear for generic complex
scattering potentials, while scattering potentials
that involve bound states in the continuum are ex-
tremely rare. For example, scattering potentials
with a compact support cannot have a bound state
in the continuum, whereas a complex potential with
a compact support can easily admit a spectral sin-
gularity [5, 9–15].

The above discussion shows that although both the
bound states in the continuum and spectral singu-
larities can be viewed as zero-width resonances, they
are quite different in nature. Most notably, spectral
singularities have a much wider domain of application,
because they occur for generic complex scattering
potentials. Concrete evidence is provided by the ob-
servation that every lasing system involves a spectral
singularity. This corresponds to lasing at the thresh-
old gain [16].

Acknowledgements
This work has been supported by the Turkish Academy of
Sciences (TÜBA).

References
[1] J. von Neumann and E. Wigner, Phys. Z. 30, 465
(1929).

[2] F. H. Stillinger and D. R. Herrick, Phys. Rev. A 11,
446 (1975).

[3] B. Gazdy Phys. Lett. 61A, 89 (1977).
[4] H. Friedrich and D. Wintgen, Phys. Rev. A 31, 3964
(1985) and 32, 3231 (1985).

[5] A. Mostafazadeh, Phys. Rev. Lett. 102, 220402 (2009).
[6] M. A. Naimark, Trudy Moscov. Mat. Obsc. 3, 181
(1954) in Russian, English translation: Amer. Math.
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[7] G. Sh. Guseinov, Pramana. J. Phys. 73, 587 (2009).
[8] I. Rotter, “The role of exceptional points in quantum
systems,” preprint arXiv: 1011.0645.

[9] A. Mostafazadeh and H. Mehri-Dehnavi, J. Phys. A
42, 125303 (2009).

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vol. 53 no. 3/2013 Spectral Singularities do not Correspond to Bound States in the Continuum

[10] A. Mostafazadeh, Phys. Rev. A 80, 032711 (2009).
[11] A. Mostafazadeh, Phys. Rev. A 84, 023809 (2011).
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[16] A. Mostafazadeh, Phys. Rev. A 83, 045801 (2011).

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	Acta Polytechnica 53(3):306–307, 2013
	Acknowledgements
	References