Spectral singularities have been extensively studied over the last one and half decades for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities has not been studied for the case of quaternionic potential. In the present work, we perform an analytical study on scattering from a quaternionic point interaction represented by a delta function. New features of spectral singularities are observed, which are different than the case of a complex (non-quaternionic) point interaction. The most notable difference is the occurrence of spectral singularity from the lossy point interaction, which is forbidden in the case of standard non-Hermitian quantum mechanics.

1.
G.
Birkhoff
and
J. V.
Neumann
,
Ann. Math.
37
,
823
(
1936
).
2.
J. V.
Neumann
,
Mathematical Foundations of Quantum Mechanics
(
Princeton University Press
,
Princeton
,
1955
).
3.
D.
Dast
,
D.
Haag
,
H.
Cartarius
,
J.
Mainand
, and
G.
Wunne
,
J. Phys. A: Math.Theor.
46
,
375301
(
2013
).
4.
D.
Dast
,
D.
Haag
,
H.
Cartarius
,
G.
Wunner
,
R.
Eichler
, and
J.
Main
,
Fortschr. Phys.
61
,
124
(
2013
).
5.
B.
Bagchi
and
A.
Banerjee
,
J. Phys. A: Math.Theor.
48
,
505201
(
2015
).
6.
J.
Cen
and
A.
Fring
,
J. Nonlinear Math. Phys.
27
,
17
(
2020
).
7.
8.
A. J.
Davies
and
B. H. J.
McKellar
,
Phys. Rev. A
46
,
3671
(
1992
).
9.
H.
Kaiser
 et al,
Phys. Rev. A
29
,
2276
(
1984
).
10.
S. L.
Adler
and
S.
Quaternionic
,
Quantum Mechanics and Quantum Fields
(
Oxford University Press
,
1995
).
11.
12.
L. M.
Procopio
 et al,
Nat. Commun.
8
,
15044
(
2017
).
13.
S. L.
Adler
,
Phys. Rev. A
95
,
060101
(
2017
).
14.
L. M.
Procopio
 et al,
Phys. Rev. A
96
,
036101
(
2017
).
15.
H.
Sobhani
 et al,
Can. J. Phys.
94
,
262
266
(
2016
).
16.
H.
Hassanabadi
 et al,
Eur. Phys. J. C
77
,
581
(
2017
).
17.
H.
Hassanabadi
 et al,
Ind. J. Phys.
91
,
1103
(
2017
).
18.
S. D.
Leo
 et al,
J. Math. Phys.
51
,
113504
(
2010
).
19.
A. J.
Davies
 et al,
Phys. Rev. A
40
,
4209
(
1989
).
20.
S. D.
Leo
 et al,
J. Math. Phys.
55
,
022301
(
2014
).
21.
S. D.
Leo
 et al,
J. Math. Phys.
47
,
082106
(
2006
).
22.
S.
Giardino
,
Adv. Appl. Clifford Algebras
27
,
2457
2471
(
2017
).
23.
S.
Giardino
,
Adv. Appl. Clifford Algebras
28
,
19
(
2018
).
24.
C. M.
Bender
and
S.
Boettcher
,
Phys. Rev. Lett.
80
,
5243
(
1998
).
25.
A.
Mostafazadeh
,
Int. J. Geom. Methods Mod. Phys.
7
,
1191
(
2010
), and references therein.
26.
C. M.
Bender
,
Rep. Prog. Phys.
70
,
947
(
2007
), and references therein.
27.
T.
Kato
,
Perturbation Theory of Linear Operators
(
Springer
,
Berlin
,
1966
).
30.
A.
Mostafazadeh
,
Phys. Rev. Lett.
102
,
220402
(
2009
).
31.
A.
Mostafazadeh
and
M.
Sarisaman
,
Phys. Lett. A
375
,
3387
(
2011
).
32.
A.
Ghatak
,
R. D.
Ray Mandal
, and
B. P.
Mandal
,
Ann. Phys.
336
,
540
(
2013
).
33.
A.
Ghatak
,
J. A.
Nathan
,
B. P.
Mandal
, and
Z.
Ahmed
,
J. Phys. A: Math. Theor.
45
,
465305
(
2012
).
34.
B. P.
Mandal
and
A.
Ghatak
,
J. Phys. A: Math Gen.
45
,
444022
(
2012
).
35.
S.
Longhi
,
J. Phys. A: Math. Theor.
44
,
485302
(
2011
).
36.
A.
Mostafazadeh
,
Phys. Rev. A
87
,
012103
(
2013
).
38.
A.
Ghatak
,
B. K.
Mourya
,
R. D.
Raymandal
, and
B. P.
Mandal
,
Int. J. Theor. Phys.
54
,
3945
(
2015
).
39.
L.
Deak
and
T.
Fulop
,
Ann. Phys.
327
,
1050
(
2012
).
42.
W.
Wan
,
Y.
Chong
,
L.
Ge
,
H.
Noh
,
A. D.
Stone
, and
H.
Cao
,
Science
331
,
889
(
2011
).
43.
N.
Liu
,
M.
Mesch
,
T.
Weiss
,
M.
Hentschel
, and
H.
Giessen
,
Nano Lett.
10
,
2342
(
2010
).
44.
H.
Noh
,
Y.
Chong
,
A.
Douglas Stone
, and
H.
Cao
,
Phys. Rev. Lett.
108
,
186805
(
2011
).
45.
A.
Mostafazadeh
and
M.
Sarisaman
,
Proc. R. Soc. A
468
,
3224
(
2012
).
46.
47.
S.
Dutta-Gupta
,
R.
Deshmukh
,
A.
Venu Gopal
,
O. J. F.
Martin
, and
S.
Dutta Gupta
,
Opt. Lett.
37
,
4452
(
2012
).
48.
A.
Ghatak
,
M.
Hasan
, and
B. P.
Mandal
,
Phys. Lett. A
379
,
1326
(
2015
).
49.
M.
Cai
,
O.
Painter
, and
K. J.
Vahala
,
Phys. Rev. Lett.
85
,
74
(
2000
).
50.
J. R.
Tischler
,
M. S.
Bradley
, and
V.
Bulovic
,
Opt. Lett.
31
,
2045
(
2006
).
51.
S.
Dutta Gupta
,
Opt. Lett.
32
,
1483
(
2007
).
52.
S.
Balci
,
C.
Kocabas
, and
A.
Aydinli
,
Opt. Lett.
36
,
2770
(
2011
).
53.
S.
Balci
,
Er.
Karademir
,
C.
Kocabas
, and
A.
Aydinli
,
Opt. Lett.
36
,
3041
(
2011
).
54.
M.
Hasan
,
A.
Ghatak
, and
B. P.
Mandal
,
Ann. Phys.
344
,
17
(
2014
).
55.
A.
Mostafazadeh
, and
M.
Sarisaman
Ann. Phys.
375
,
265
(
2016
).
56.
Z. H.
Musslimani
,
K. G.
Makris
,
R.
El-Ganainy
, and
D. N.
Christodoulides
,
Phys. Rev. Lett.
100
,
030402
(
2008
).
57.
C. E.
Ruter
,
K. G.
Makris
,
R.
El-Ganainy
,
D. N.
Christodoulides
,
M.
Segev
, and
D.
Kip
,
Nat. Phys.
6
,
192
(
2010
).
58.
R.
El-Ganainy
,
K. G.
Makris
,
D. N.
Christodoulides
, and
Z. H.
Musslimani
,
Opt. Lett.
32
,
2632
(
2007
).
59.
A.
Guo
 et al,
Phys. Rev. Lett.
103
,
093902
(
2009
).
60.
M.
Hasan
and
B. P.
Mandal
,
Ann. Phys.
396
,
371
(
2018
).
61.
A.
Mostafazadeh
, in
Scattering Theory and PT-Symmetry in Parity-Time Symmetry and its Applications
, edited by
D.
Christodoulides
and
J.
Yang
(
Springer
,
2018
).
62.
A.
Mostafazadeh
,
Phys. Rev. A
80
,
032711
(
2009
).
63.
A.
Mostafazadeh
,
J. Phys. A: Math. Gen.
39
,
43
(
2006
).
You do not currently have access to this content.