Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum

Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following pecul...

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Datum:	2011
Hauptverfasser:	Andrianov, A.A., Sokolov, A.V.
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spelling	irk-123456789-1480882019-02-17T01:26:18Z Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum Andrianov, A.A. Sokolov, A.V. Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined. 2011 Article Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum / A.A. Andrianov, A.V. Sokolov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81Q60; 81R15; 47B15 DOI: http://dx.doi.org/10.3842/SIGMA.2011.111 http://dspace.nbuv.gov.ua/handle/123456789/148088 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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description	Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined.
format	Article
author	Andrianov, A.A. Sokolov, A.V.
spellingShingle	Andrianov, A.A. Sokolov, A.V. Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Andrianov, A.A. Sokolov, A.V.
author_sort	Andrianov, A.A.
title	Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
title_short	Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
title_full	Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
title_fullStr	Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
title_full_unstemmed	Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
title_sort	resolutions of identity for some non-hermitian hamiltonians. i. exceptional point in continuous spectrum
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/148088
citation_txt	Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum / A.A. Andrianov, A.V. Sokolov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT andrianovaa resolutionsofidentityforsomenonhermitianhamiltoniansiexceptionalpointincontinuousspectrum AT sokolovav resolutionsofidentityforsomenonhermitianhamiltoniansiexceptionalpointincontinuousspectrum
first_indexed	2025-07-12T18:12:26Z
last_indexed	2025-07-12T18:12:26Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 111, 19 pages Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum? Alexander A. ANDRIANOV †‡ and Andrey V. SOKOLOV † † V.A. Fock Department of Theoretical Physics, Sankt-Petersburg State University, 198504 St. Petersburg, Russia E-mail: andrianov@bo.infn.it, avs avs@rambler.ru ‡ ICCUB, Universitat de Barcelona, 08028 Barcelona, Spain E-mail: andrianov@icc.ub.edu Received August 06, 2011, in final form November 25, 2011; Published online December 05, 2011 http://dx.doi.org/10.3842/SIGMA.2011.111 Abstract. Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the contin- uous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined. Key words: non-Hermitian quantum mechanics; supersymmetry; exceptional points; resolu- tion of identity 2010 Mathematics Subject Classification: 81Q60; 81R15; 47B15 1 Introduction The interest to exceptional points in non-Hermitian quantum dynamical systems has been re- voked recently [1, 2, 3, 4, 5] although their very notion exists quite a time [6, 7, 8, 9, 10]. Their appearance can be associated to level coalescence at complex coupling constants for initially Hermitian Hamiltonians [11]. If existing they play an important role in definition of energy spectra and in construction of biorthogonal bases in Riesz spaces [12]. Whereas the appearance of exceptional points in discrete spectrum does not give rise to any principal obstacles for building a resolution of identity, the emergence of exceptional points inside or on the border of continuous spectrum makes the very construction of resolutions of identity rather sophisticated [2]. Their correct description in brief represents the main aim of our paper (entitled as Part I) whereas in the subsequent paper [13] (entitled as Part II) the detailed proofs of the results announced here are presented by one of us (A.V.S.). Let us start with the notion of exceptional point and further on outline the structure of the present work. The spectrum of a Hermitian Hamiltonian, in general, consists of continuous part ?This paper is a contribution to the Proceedings of the Workshop “Supersymmetric Quantum Me- chanics and Spectral Design” (July 18–30, 2010, Benasque, Spain). The full collection is available at http://www.emis.de/journals/SIGMA/SUSYQM2010.html mailto:andrianov@bo.infn.it mailto:avs_avs@rambler.ru mailto:andrianov@icc.ub.edu http://dx.doi.org/10.3842/SIGMA.2011.111 http://www.emis.de/journals/SIGMA/SUSYQM2010.html 2 A.A. Andrianov and A.V. Sokolov and discrete points. Meanwhile the spectrum of a non-Hermitian Hamiltonian may contain also a new type of spectral points embedded into a continuous spectrum and/or into a discrete one, namely, exceptional points. The exceptional point of the spectrum of one-dimensional Hamiltonian h defined on entire axis is an eigenvalue λ0 of this Hamiltonian for which there is a normalizable eigenfunction ψ0(x) and also a number of associated functions [7] ψj(x), j = 1, . . . , n− 1: hψ0 = λ0ψ0, (h− λ0)ψj = ψj−1, j = 1, . . . , n− 1. For a discrete spectrum the latter ones are typically normalizable [2, 3]. On the other hand some non-normalizable associated functions bounded or even growing at infinity may be involved in building of resolution of identity as well. It will be proven in Part II. The number n is an algebraic multiplicity of λ0. Thus, in continuous spectrum one can deal with two types of algebraic multiplicities (which are not necessarily equal: their different types for continuous spectrum are discussed in conclusions). If an exceptional point λ0 belongs to a discrete part of the spectrum then n simultaneously characterizes the order of a pole at E = λ0 of the Green function. For an exceptional point λ0 on the border of continuous spectrum the Green function reveals (see Section 2.3) a branching point with the pole order 2n+ 1 in the variable √ E − λ0, where n is a maximal number of linearly independent eigen- and formal associated functions of h for an eigenvalue λ0 in the resolution of identity. When an exceptional point lies inside of continuous part of the spectrum the pole order may be larger than n (which has the same meaning as in the previous sentence) that is elucidated in details in conclusions. In this paper we build resolutions of identity for certain non-Hermitian Hamiltonians con- structed from biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with contin- uous spectrum and the following peculiarities are investigated: in Section 2 the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; in Section 3 the case when there is an exceptional point inside of continuous spectrum. In Sec- tion 4 in conclusions the different ways to introduce algebraic multiplicities are discussed and the SUSY tools [14, 15, 16, 17, 18, 19, 20, 21, 22] in regulating them are inspected. More specifically in Sections 2 and 3 the reductions of the resolutions of identity under narrowing of the classes of employed test functions are elaborated. It is shown that the bounded associated function in an exceptional point inside of continuous spectrum does not belong to the physical state space (i.e. does not belong to the complete biorthogonal system built from eigenfunctions of the Hamiltonian and cannot be reproduced with the help of harmonic expansion generated by an appropriate resolution of identity). If an exceptional point lies on a boundary of continuous spectrum then some of associated functions included into the resolution of identity are normalizable and some of them may be not, still being elements of a rigged Hilbert space [23] and its dual one, the Gelfand triple generalized onto biorthogonal resolutions of identity. 2 Resolutions of identity for model Hamiltonians with an exceptional point of arbitrary multiplicity at the bottom of continuous spectrum 2.1 Basic constructions Let us consider the sequence of Hamiltonians, hn = −∂2 + n(n+ 1) (x− z)2 , x ∈ R, ∂ ≡ d dx , Im z 6= 0, n = 0, 1, 2, . . . , Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 3 where h0 is the Hamiltonian of a free particle and all these Hamiltonians are PT -symmetric [24] for the choice Re z = 0. One can easily check that for the Hamiltonian hn on the energy level E = 0 there is an eigenfunction ψn0(x) and a chain of formal associated functions ψnl(x): hnψn0 = 0, hnψnl = ψn,l−1, l = 1, 2, 3, . . . , ψn0(x) = (−i)n(2n− 1)!!√ 2π (x− z)n , ψnl(x) = (−i)n(2n− 2l − 1)!!√ 2π(2l)!!(x− z)n−2l , l = 0, 1, 2, . . . , 0!! = (−1)!! = 1, (−2m− 1)!! = (−1)m (2m− 1)!! , m = 1, 2, 3, . . . . (2.1) Moreover for odd n the functions ψnl(x); l = 0, . . . , [n/2] are normalizable (i.e. belong to L2(R)) and when l > [n/2] they are non-normalizable and unboundedly growing for x→ ±∞. For even n the functions ψnl(x); l = 0, . . . , [n/2]− 1 are normalizable, the function ψn,n/2(x) ≡ (−i)n(n− 1)!!√ 2π n!! (2.2) is bounded but non-normalizable and the functions ψnl(x) for l > n/2 are non-normalizable and unboundedly growing for x→ ±∞. The Hamiltonians hn, n = 0, 1, 2, . . . are intertwined by the operators q±n = ∓∂ − χn(x), χn(x) = ψ′n0(x) ψn0(x) ≡ − n x− z , q−n ψn0 = 0, n = 0, 1, 2, . . . , (2.3) with the help of the chain (ladder) construction [25, 26, 27, 17] hnq + n = q+n hn−1, q−n hn = hn−1q − n and hn = q+n q − n , hn−1 = q−n q + n , n = 0, 1, 2, . . . , h−1 = −∂2 = h0. One easily check that q−n ψnl = −iψn−1,l−1, n, l = 1, 2, 3, . . . . (2.4) As well the eigenfunctions ψn(x; k) for continuous spectrum of the Hamiltonian hn can be produced from the eigenfunctions ψ0(x; k) = 1√ 2π eikx, k ∈ R for the continuous spectrum of the Hamiltonian h0 of a free particle with the help of intertwining operators (2.3): ψn(x; k) = 1√ 2π ( i k )n q+n · · · q+1 e ikx, hnψn(x; k) = k2ψn(x; k), k ∈ (−∞, 0) ∩ (0,+∞), n = 0, 1, 2, . . . . (2.5) For the function ψn(x; k) one can derive the following explicit representation by induction, ψn(x; k) = 1√ 2π eikx n∑ m=0 (n+m)! 2mm!(n−m)! im km(x− z)m , (2.6) 4 A.A. Andrianov and A.V. Sokolov where from it follows that the cofactor (i/k)n in (2.5) provides the asymptotic form for ψn(x; k) = 1√ 2π eikx [ 1 +O ( 1 x )] , x→ ±∞. (2.7) As well, using (2.3) and (2.5), we can get the following representations for ψn(x; k): ψn(x; k) = 1√ 2π eikz ( d dt − n t )( d dt − n− 1 t ) · · · ( d dt − 1 t ) et ∣∣∣∣ t=ik(x−z) = eikz√ 2πin(x− z)n ( ∂ ∂k − n k )( ∂ ∂k − n− 1 k ) · · · ( ∂ ∂k − 1 k ) eik(x−z). (2.8) At last, it follows from (2.5) and (2.8) that ψn(x; k) = i k q+n ψn−1(x; k) = eikz i(x− z) ( ∂ ∂k − n k )[ e−ikzψn−1(x; z) ] , n = 1, 2, 3, . . . . (2.9) Let us find now the connection between ψn(x; k) and ψnl(x), l = 0, 1, 2, . . . . For this purpose we calculate the following derivative, using Leibnitz formula and the relation 4.2.7.14 from [28], lim k→0 ∂m ∂km [ e−ikzknψn(x; k) ] = ∂m ∂km [ 1√ 2π eik(x−z) n∑ s=0 (n+ s)! 2ss!(n− s)! iskn−s (x− z)s ] ∣∣∣ k=0 = in+mn!√ 2π 2n(x− z)n−m min{m,n}∑ j=0 (−2)jCjmCn2n−j = in+m√ 2π (x− z)n−m n∏ s=1 (2s−m− 1) =  (−1)nm!ψn,m/2(x), m is even, 0, m is odd and 6 2n− 1, im−n(m− 1)!!√ 2π(m− 2n− 1)!! 1 (x− z)n−m , m is odd and > 2n− 1, (2.10) where Cmn ≡ n!/[m!(n−m)!] is a binomial coefficient. Thus, ψnl(x) = (−1)n (2l)! lim k→0 ∂2l ∂k2l [ e−ikzknψn(x; k) ] , l = 0, 1, 2, . . . . (2.11) Since the continuous spectrum of the Hamiltonian hn, n = 1, 2, 3, . . . coincide with [0,+∞) (see (2.5)), the eigenvalue E = 0 of this Hamiltonian for the chain of functions ψnl(x), l = 0, 1, 2, . . . is situated at the bottom of continuous spectrum. It will be shown in Section 2.3 which of these functions are included in the resolution of identity constructed from eigen- and associated functions of hn. 2.2 Biorthogonality relations The biorthogonality relations between functions ψnl(x), l = 0, . . . , n− 1 follow from (2.1):∫ +∞ −∞ ψnl(x)ψnl′(x) dx = 0, l + l′ 6 n− 1. (2.12) The biorthogonality relations between ψn0(x) and ψn(x; k) can be derived with the help of (2.3), (2.7) and (2.9):∫ +∞ −∞ ψn0(x) [ knψn(x; k) ] dx = i ∫ +∞ −∞ ψn0(x) ( −∂ + n x− z ) [kn−1ψn−1(x; k)] dx Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 5 = −iψn0(x) [ kn−1ψn−1(x; k) ]∣∣∣+∞ −∞ + i ∫ +∞ −∞ [q−n ψn0(x)] [ kn−1ψn−1(x; k) ] dx = 0, n = 1, 2, 3, . . . . (2.13) The biorthogonality relations between normalizable associated functions ψnl(x) and ψn(x; k) can be derived in the same way with the help of (2.4) by induction,∫ +∞ −∞ ψnl(x)[k nψn(x; k)] dx = i ∫ +∞ −∞ ψnl(x) ( −∂ + n x− z )[ kn−1ψn−1(x; k) ] dx = −iψnl(x)[kn−1ψn−1(x; k)] ∣∣∣+∞ −∞ + i ∫ +∞ −∞ [q−n ψnl(x)] [ kn−1ψn−1(x; k) ] dx = ∫ +∞ −∞ ψn−1,l−1(x) [ kn−1ψn−1(x; k) ] dx = · · · = ∫ +∞ −∞ ψn−l,0(x) [ kn−lψn−l(x; k) ] dx = 0, l = 1, . . . , [ n− 1 2 ] . (2.14) The biorthogonality relations between non-normalizable formal associated functions ψnl(x), l = [(n+ 1)/2], . . . , n− 1 and ψn(x; k),∫ +∞ −∞ ψnl(x) [ knψn(x; k) ] dx = 0, l = [ n+ 1 2 ] , . . . , n− 1, (2.15) can be derived with the help of (2.1), (2.6), the Jordan lemma and the relation 4.2.7.17 from [28] as follows,∫ +∞ −∞ ψnl(x)[k nψn(x; k)] dx = (−i)n (2n− 2l − 1)!! 2π(2l)!! 2l−n∑ m=0 im(n+m)! 2mm!(n−m)! kn−m ∫ +∞ −∞ (x− z)2l−n−meikx dx + (−i)n (2n− 2l − 1)!! 2π(2l)!! n∑ m=2l−n+1 im(n+m)! 2mm!(n−m)! kn−m ∫ +∞ −∞ eikx dx (x− z)n−2l+m = (−i)n (2n− 2l − 1)!! (2l)!! 2l−n∑ m=0 im(n+m)! 2mm!(n−m)! kn−m ( −i d dk − z )2l−n−m δ(k) + sign (Im z)θ(sign (Im z) k)(−i)n (2n− 2l − 1)!! (2l)!! eikz × n∑ m=2l−n+1 in−2l+2m(n+m)! 2mm!(n−m)!(n− 2l +m− 1)! k2n−2l−1 = (−i)n (2n− 2l − 1)!! (2l)!! 2l−n∑ m=0 im(n+m)! 2mm!(n−m)! kn−m ( −i d dk − z )(n−m)−2(n−l) δ(k) + sign (Im z)θ(sign (Im z) k)(−1)l2n (2n− 2l − 1)!!(2l + 1)!! (2n)!! k2n−2l−1eikz × n∑ m=2l−n+1 (−1)m 2m Cmn C 2l+1 m+n = 0, where θ(t) = { 1, t > 0, 0, t < 0. 6 A.A. Andrianov and A.V. Sokolov The formal associated functions ψnl(x), l = n, n + 1, n + 2, . . . are not contained in the resolutions of identity (see Section 2.3), but it is interesting that one can write the biorthogonality relations for these functions with ψn(x; k) as well,∫ +∞ −∞ ψnl(x) [ e−ikzknψn(x; k) ] dx = (−1)lin 2π(2l)!!(2l − 2n− 1)!! e−ikz n∑ m=0 im(n+m)! 2mm!(n−m)! kn−m ∫ +∞ −∞ (x− z)2l−n−meikx dx = (−1)lin (2l)!!(2l − 2n− 1)!! e−ikz n∑ m=0 im(n+m)! 2mm!(n−m)! kn−m ( −i d dk − z )2l−n−m δ(k) = (−1)n (2l)!!(2l − 2n− 1)!! e−ikz n∑ m=0 (−1)m(n+m)! 2mm!(n−m)! kn−m ( d dk − iz )2l−n−m [eikzδ(k)] = (−1)n (2l)!!(2l − 2n− 1)!! n∑ m=0 (−1)m(n+m)! 2mm!(n−m)! kn−mδ(2l−n−m)(k) = 1 (2l)!!(2l − 2n− 1)!!(2l − 2n)! δ(2l−2n)(k) n∑ m=0 (n+m)!(2l − n−m)! 2mm!(n−m)! = 1 (2l − 2n)! δ(2l−2n)(k), l = n, n+ 1, n+ 2, . . . , (2.16) where we have used (2.1), (2.6) and the relation n∑ m=0 (n+m)!(s−m)! 2mm!(n−m)! = n∑ m=0 (n−m+ 2m)(n− 1 +m)!(s−m)! 2mm!(n−m)! = n−1∑ m=0 (n− 1 +m)!(s−m)! 2mm!(n− 1−m)! + n∑ m=1 (n− 1 +m)!(s−m)! 2m−1(m− 1)!(n−m)! = n−1∑ m=0 (n− 1 +m)!(s−m)! 2mm!(n− 1−m)! + n−1∑ m=0 (n+m)!(s− 1−m)! 2mm!(n− 1−m)! = (s+ n) n−1∑ m=0 (n− 1 +m)!(s− 1−m)! 2mm!(n− 1−m)! = · · · = (s+ n) · · · (s− n+ 2)(s− n)! = (s− n− 1)!!(s+ n)!!, s > n. At last, the biorthogonality relations between eigenfunctions for continuous spectrum of the Hamiltonian hn are proved in [13] and take the following form:∫ +∞ −∞ [knψn(x; k)][(k ′)nψn(x;−k′)] dx = (k′)2nδ(k − k′). (2.17) Let us notice that (2.13)–(2.15) contain (2.12) for l = 0, . . . , n−1, l′ = 0 and (2.17) contains (2.12) for l = l′ = 0 due to (2.11). The relations (2.16) can be derived with the help of differentiation from (2.17) also in view of (2.11). 2.3 Resolutions of identity The initial resolution of identity constructed from ψn(x; k) holds [13], δ(x− x′) = ∫ L ψn(x; k)ψn(x ′;−k) dk, (2.18) Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 7 where L is an integration path in complex k plane, obtained from the real axis by its deformation near the point k = 0 upwards or downwards (the direction of this deformation is of no difference since the residue of the integrand for the point k = 0 is equal zero in view of (2.6) and (2.10)) and the direction of L is specified from −∞ to +∞. This resolution of identity is valid for test functions belonging to CLγ ≡ C∞R ∩ L2(R; (1 + \|x\|)γ), γ > −1 as well as for some bounded and even slowly increasing test functions (more details are presented in [13]) and, in particular, for eigenfunctions ψn(x; k) and for the associated function (2.2). One can rearrange the resolution of identity (2.18) for any ε > 0 to the forms δ(x− x′) = (∫ −ε −∞ + ∫ +∞ ε ) ψn(x; k)ψn(x ′;−k) dk + n−1∑ l=0 ( x′ − z x− z )l ψn−l−1(x; k)ψn−l(x′;−k) i(x− z) ∣∣∣ε −ε + (x′ − z x− z )n sin ε(x− x′) π(x− x′) ≡ (∫ −ε −∞ + ∫ +∞ ε ) ψn(x; k)ψn(x ′;−k) dk + sin ε(x− x′) π(x− x′) − cos ε(x− x′) 2πε(x− z)(x′ − z) n−1∑ l=0 (−1/4)l ε2l(x′ − z)2l min{2l,n−1}∑ m=0 C2l+1,m,n (n+2l+1−m)! (n− 1−m)! ( x′ − z x− z )m + sin ε(x− x′) π(x− z) n−1∑ l=1 (−1/4)l ε2l(x′ − z)2l min{2l−1,n−1}∑ m=0 C2l,m,n (n+ 2l −m)! (n− 1−m)! ( x′ − z x− z )m , (2.19) Clmn = 1 l m∑ j=0 (−1)jCjl C l−1 n−m−1+2j , n = 1, 2, 3, . . . , l = 1, . . . , 2n− 1, m = 0, . . . ,min{l − 1, n− 1} (cf. with (68) in [2]) and, consequently, to the form δ(x− x′) = lim ε↓0 ′ {(∫ −ε −∞ + ∫ +∞ ε ) ψn(x; k)ψn(x ′;−k) dk + sin ε(x− x′) π(x− x′) − cos ε(x− x′) 2πε(x− z)(x′ − z) n−1∑ l=0 (−1/4)l ε2l(x′ − z)2l min{2l,n−1}∑ m=0 C2l+1,m,n (n+2l+1−m)! (n− 1−m)! ( x′ − z x− z )m + sin ε(x− x′) π(x− z) n−1∑ l=1 (−1/4)l ε2l(x′ − z)2l min{2l−1,n−1}∑ m=0 C2l,m,n (n+ 2l −m)! (n− 1−m)! ( x′ − z x− z )m} , (2.20) where the prime ′ at the limit symbol emphasizes that this limit is regarded as a limit in the space of distributions (see details in [13]). From (2.19) and (2.20) one can try to derive various reduced resolutions of identity similar to the resolutions (69) and (70) of [2], which correspond to the partial case n = 1, or to the resolutions (15) and (16) of [29]. In particular, by virtue of Lemma 3.7 from [13], for test functions from CLγ , γ > −1 the resolution (2.20) can be reduced to the form δ(x− x′) = lim ε↓0 ′ {(∫ −ε −∞ + ∫ +∞ ε ) ψn(x; k)ψn(x ′;−k) dk − cos ε(x− x′) 2πε(x− z)(x′ − z) n−1∑ l=0 (−1/4)l ε2l(x′ − z)2l min{2l,n−1}∑ m=0 C2l+1,m,n (n+2l+1−m)! (n− 1−m)! ( x′ − z x− z )m 8 A.A. Andrianov and A.V. Sokolov + sin ε(x− x′) π(x− z) n−1∑ l=1 (−1/4)l ε2l(x′ − z)2l min{2l−1,n−1}∑ m=0 C2l,m,n (n+ 2l −m)! (n− 1−m)! ( x′ − z x− z )m} .(2.21) More reduced resolutions of identity for the partial case n = 2 are presented in the forthcoming Section 2.4. There is another way to transform the resolution (2.18) as well. This way was used for obtaining of the resolution of identity (26) from [29]. The integral from the right-hand part of (2.18) is understood [13] as follows:∫ L ψn(x; k)ψn(x ′;−k) dk = lim′ A→+∞ ∫ L(A) ψn(x; k)ψn(x ′;−k) dk, (2.22) where L(A) is a path in complex k plane, made of the segment [−A,A] by its deformation near the point k = 0 upwards or downwards (the direction of this deformation is of no difference as well as in the case with L) and the direction of L(A) is specified from −A to A. When using the facts that the integral in the right-hand part of (2.22) is a standard integral (not a distribution) and that k2nψn(x; k)ψn(x ′;−k) is an entire function of k (see (2.6)) as well as employing the Leibniz formula and the formulae (2.10) and (2.11), we can transform (2.22) as follows,∫ L ψn(x; k)ψn(x ′;−k) dk = lim′ A→+∞ lim ε↓0 {(∫ −ε −A + ∫ A ε ) ψn(x; k)ψn(x ′;−k) dk + 2n−1∑ j=0 1 j! ∂j ∂kj [ k2nψn(x; k)ψn(x ′;−k) ]∣∣∣ k=0 ∫ L(ε) dk k2n−j } = lim′ A→+∞ lim ε↓0 {(∫ −ε −A + ∫ A ε ) ψn(x; k)ψn(x ′;−k) dk − 2(−1)n n−1∑ l=0 1 (2n− 2l − 1)ε2n−2l−1 l∑ m=0 ψnm(x)ψn,l−m(x ′) } = lim∗ ε↓0 {(∫ −ε −∞ + ∫ +∞ ε ) ψn(x; k)ψn(x ′;−k) dk − 2(−1)n n−1∑ l=0 1 (2n− 2l − 1)ε2n−2l−1 l∑ m=0 ψnm(x)ψn,l−m(x ′) } , (2.23) where the latter equality is considered as a definition and the limit lim ε↓0 is regarded as point-wise one (not as a limit in a function space). Let us show that the terms outside the integral in the last line of (2.23) can be rearranged as follows, −2(−1)n n−1∑ l=0 1 (2n− 2l − 1)ε2n−2l−1 l∑ m=0 ψnm(x)ψn,l−m(x ′) = n−1∑ l=0 ψnl(x; ε)ψn,n−1−l(x ′; ε), (2.24) where ψnl(x; ε), l = 0, . . . , n − 1 is the chain of eigenfunction and associated functions (formal for l = [(n+ 1)/2], . . . , n− 1) of the Hamiltonian hn for the eigenvalue E = 0 of the form ψnl(x; ε) = l∑ j=0 αj(ε)ψn,l−j(x), l = 0, . . . , n− 1, (2.25) Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 9 hnψn0(x; ε) = 0, hnψnl(x; ε) = ψn,l−1(x; ε), l = 1, . . . , n− 1 with αj(ε), j = 0, . . . , n− 1 being unknown coefficients which will be found below. Using (2.24) and (2.25), it is easy to check that (2.24) is valid iff the coefficients αj(x) satisfy the following system, l∑ j=0 αj(ε)αl−j(ε) = − 2(−1)n (2l + 1)ε2l+1 , l = 0, . . . , n− 1. (2.26) After the redefinition αj(ε) = √ 2 in+1 ε2j √ ε βj , j = 0, . . . , n− 1, where βj , j = 0, . . . , n− 1 are new unknown coefficients, the system (2.26) takes the form l∑ j=0 βjβl−j = 1 2l + 1 , l = 0, . . . , n− 1. The general solution of the latter system can be found in the recurrent form, β0 = ±1, β1 = 1 6β0 , βl = 1 2β0  1 2l + 1 − l−1∑ j=1 βjβl−j  , l = 2, . . . , n− 1. The first terms of the sequence βj , j = 0, . . . , n− 1 in the case β0 = 1 are the following ones, β0 = 1, β1 = 1 6 , β2 = 31 360 , β3 = 863 15120 , β4 = 76813 1814400 , . . . . Thus, we can choose the functions ψnl(x; ε) in the form, ψnl(x; ε) = in+1 √ 2 ε l∑ j=0 βj ε2j ψn,l−j(x), l = 0, . . . , n− 1 (2.27) and the resolution of identity holds, δ(x− x′) = lim∗ ε↓0 {(∫ −ε −∞ + ∫ +∞ ε ) ψn(x; k)ψn(x ′;−k) dk + n−1∑ l=0 ψnl(x; ε)ψn,n−1−l(x ′; ε) } , (2.28) (cf. with (69) from [2] for the case n = 1). Moreover, this resolution is equivalent to (2.18), i.e. it is valid for all test functions for which (2.18) is valid (cf. with the analogous results in Section 3 and in [29]). The resolution of identity (2.28) contains all n functions from the chain ψnl(x; ε), l = 0, . . . , n − 1 and in this resolution for the eigenvalue E = 0 there are no other eigen- or as- sociated functions of the Hamiltonian hn. The order of the pole k = 0 for the Green function Gn(x, x ′;E) = [ πi k ψn(x>; k)ψn(x<;−k) ] ∣∣∣ k= √ E , x> = max{x, x′}, x< = {x, x′}, Im √ E > 0, (hn − E)Gn = δ(x− x′) (2.29) 10 A.A. Andrianov and A.V. Sokolov considered as a function of k = √ E is equal to 2n + 1 in view of (2.6), and the exceptional point E = 0 coincides with the branch point of this Green function as a function of E. One can consider this pole of the order 2n+ 1 as a result of confluence of the pole of the order n for the Green function as a function of E = k2 and of the factor k ≡ √ E from the denominator of the Green function (see (2.29)). Thus, the number n of linearly independent eigen- and (formal) associated functions of the Hamiltonian hn incorporated in the resolution of identity (2.28) for the eigenvalue E = 0 (exceptional point) is equal to the order of the “pole” E = 0 for the Green function Gn(x, x ′;E) in the sense elucidated above or, more rigorously, the order of the pole of Gn(x, x ′;E) as a function of k = √ E is 2n+ 1 expressed in terms of the number n. Let us notice that in view of (2.27) the functions ψnl(x; ε), l = 0, . . . , n− 1 satisfy the same biorthogonality relations from Section 2.2 as the functions ψnl(x), l = 0, . . . , n− 1. 2.4 Example: case n = 2 For the Hamiltonian h2 = −∂2 + 6 (x− z)2 , x ∈ R, Im z 6= 0, there are continuous spectrum eigenfunctions ψ2(x; k) = 1√ 2π [ 1− 3 ik(x− z) − 3 k2(x− z)2 ] eikx, h2ψ2(x; k) = k2ψ2(x; k), k ∈ (−∞, 0) ∪ (0,+∞) (2.30) (see (2.6)) and also the normalizable eigenfunction ψ20(x) and the bounded associated func- tion ψ21(x) on the level E = 0, ψ20(x) = − 3√ 2π (x− z)2 , ψ21(x) = − 1 2 √ 2π , h2ψ20 = 0, h2ψ21 = ψ20, (see (2.1) and (2.2)). In this case at the point k = 0 there is a fifth order pole in the Green function G2(x, x ′;E) considered as a function of k = √ E (see (2.29) and (2.30)) and thereby the exceptional point E = 0 of the spectrum of the Hamiltonian h2 coincides with the branch point for the Green function as a function of E. In the case under consideration the biorthogonality relations (2.12)–(2.15) and (2.17) take the form,∫ +∞ −∞ ψ2 20(x) dx = 0, ∫ +∞ −∞ ψ20(x) [ k2ψ2(x; k) ] dx = 0, (2.31)∫ +∞ −∞ ψ20(x)ψ21(x) dx = 0, ∫ +∞ −∞ ψ21(x) [ k2ψ2(x; k) ] dx = 0, (2.32)∫ +∞ −∞ [ k2ψ2(x; k) ][ (k′)2ψ2(x;−k′) ] dx = (k′)4δ(k − k′), (2.33) where (2.31) are included in (2.33) due to the equality ψ20(x) = lim k→0 [ k2ψ2(x; k) ] (2.34) (see (2.11)) and (2.32) can be derived from (2.33) in view of (2.34) and of the equality ψ21(x) = 1 2 lim k→0 ∂2 ∂k2 [ e−ikzk2ψ2(x; k) ] (see (2.11) as well). Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 11 It is straightforward to check that the resolution of identity (2.19) (which is valid for test functions from CLγ , γ > −1 as well as for some bounded and even slowly increasing test functions, see Section 2.3), can be rewritten in the form δ(x− x′) = (∫ −ε −∞ + ∫ +∞ ε ) ψ2(x; k)ψ2(x ′;−k) dk + [ ψ20(x; ε)ψ21(x ′; ε) + ψ21(x; ε)ψ20(x ′; ε) ] + sin ε(x− x′) π(x− x′) + 6 sin2 ε2(x− x ′) πε(x− z)(x′ − z) + 12(x− x′) sin2 ε4(x− x ′) sin ε 2(x− x ′) πε2(x− z)2(x′ − z)2 + 3[ε(x− x′)− 2 sin ε 2(x− x ′)]2 2πε3(x− z)2(x′ − z)2 , (2.35) where the eigenfunction ψ20(x; ε) and the associated function ψ21(x; ε) of the Hamiltonian h2 read ψ20(x; ε) = −i √ 2 ε ψ20(x) ≡ 3i√ πε (x− z)2 , ψ21(x; ε) = −i √ 2 ε [ ψ21(x) + 1 6ε2 ψ20(x) ] ≡ i 2 √ πε [ 1 + 1 ε2(x− z)2 ] , h2ψ20(x; ε) = 0, h2ψ21(x; ε) = ψ20(x; ε). The eigenfunction ψ20(x; ε) and the associated function ψ21(x; ε) obviously satisfy the biorthogo- nality relations similar to (2.31) and (2.32). It is shown in [13] that the resolution of identity (2.35) can be reduced: a) for test functions from CLγ , γ > −1 to the form δ(x− x′) = lim ε↓0 ′ {(∫ −ε −∞ + ∫ +∞ ε ) ψ2(x; k)ψ2(x ′;−k) dk + [ ψ20(x; ε)ψ21(x ′; ε) + ψ21(x; ε)ψ20(x ′; ε) ] + 6 sin2 ε2(x− x ′) πε(x− z)(x′ − z) + 12(x− x′) sin2 ε4(x− x ′) sin ε 2(x− x ′) πε2(x− z)2(x′ − z)2 + 3[ε(x− x′)− 2 sin ε 2(x− x ′)]2 2πε3(x− z)2(x′ − z)2 } (2.36) identical to (2.21), b) for test functions from CLγ , γ > 1 to the form δ(x− x′) = lim ε↓0 ′ {(∫ −ε −∞ + ∫ +∞ ε ) ψ2(x; k)ψ2(x ′;−k) dk + [ ψ20(x; ε)ψ21(x ′; ε) + ψ21(x; ε)ψ20(x ′; ε) ] + 12(x− x′) sin2 ε4(x− x ′) sin ε 2(x− x ′) πε2(x− z)2(x′ − z)2 + 3[ε(x− x′)− 2 sin ε 2(x− x ′)]2 2πε3(x− z)2(x′ − z)2 } (2.37) and c) for test functions from CLγ , γ > 3 to the form δ(x− x′) = lim ε↓0 ′ {(∫ −ε −∞ + ∫ +∞ ε ) ψ2(x; k)ψ2(x ′;−k) dk + ψ20(x; ε)ψ21(x ′; ε) + ψ21(x; ε)ψ20(x ′; ε) } . (2.38) 12 A.A. Andrianov and A.V. Sokolov The latter of these resolutions of identity seems to have a more natural form than the pre- vious ones, but the right-hand part of the latter resolution cannot reproduce the normalizable eigenfunction ψ20(x) 6∈ CLγ ≡ C∞R ∩ L2(R; (1 + \|x\|)γ), γ > 3 because of the biorthogonality relations. With the help of the Jordan lemma one can check that lim ε↓0 ∫ +∞ −∞ 12(x− x′) sin2 ε4(x− x ′) sin ε 2(x− x ′) πε2(x− z)2(x′ − z)2 ψ20(x) dx = lim ε↓0 {[ −3 4 e±iε(z−x ′)/2 ∓ i 8 ε(z − x′)e±iε(z−x′)/2 + 3 2 e±iε(z−x ′) ± i 2 ε(z − x′)e±iε(z−x′) ] ψ20(x ′) } = 3 4 ψ20(x ′) (2.39) and lim ε↓0 ∫ +∞ −∞ 3[ε(x− x′)− 2 sin ε 2(x− x ′)]2 2πε3(x− z)2(x′ − z)2 ψ20(x) dx (2.40) = lim ε↓0 {[ 3 4 e±iε(z−x ′)/2 ± i 8 ε(z − x′)e±iε(z−x′)/2 − 1 2 e±iε(z−x ′) ] ψ20(x ′) } = 1 4 ψ20(x ′), where the upper (lower) signs correspond to the case Im z > 0 (Im z < 0). Hence, just two last terms of the resolution of identity (2.37) and the corresponding terms in the resolutions of identity (2.35) and (2.36) give a chance to reproduce ψ20(x) by these resolutions (cf. with the analogous results in Section 6.1 of [2], in Section 2 of [29] and in Section 3 of the present paper). It is interesting that contributions of these terms in the resolution of identity are (see Remark 3.4 in [13]) singular discontinuous functionals whose supports consist of the only element which is the infinity (cf. with the analogous comments in Section 2 of [29] and in Section 3 of the present paper). In the case under consideration the resolution of identity (2.28) takes the following form, δ(x− x′) = lim ε↓0 ∗ {(∫ −ε −∞ + ∫ +∞ ε ) ψ2(x; k)ψ2(x ′;−k) dk + ψ20(x; ε)ψ21(x ′; ε) + ψ21(x; ε)ψ20(x ′; ε) } , (cf. with (2.38)) and it is valid for test functions from CLγ , γ > −1 as well as for some bounded and even slowly increasing test functions (see Section 2.3). 3 Resolutions of identity for the model Hamiltonian with exceptional point inside of the continuous spectrum For the Hamiltonian h = −∂2 + 16α2α(x− z) sin 2αx+ 2 cos2 αx [sin 2αx+ 2α(x− z)]2 , x ∈ R, ∂ ≡ d dx , α > 0, Im z 6= 0 Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 13 there are [2] continuous spectrum eigenfunctions ψ(x; k) = 1√ 2π [ 1 + ik k2 − α2 W ′(x) W (x) − 1 2(k2 − α2) W ′′(x) W (x) ] eikx, W (x) = sin 2αx+ 2α(x− z), hψ(x; k) = k2ψ(x; k), k ∈ (−∞,−α) ∪ (−α, α) ∪ (α,+∞). (3.1) As well for the level E = α2 there is the normalizable eigenfunction ψ0(x) and the bounded associated function1 ψ1(x), ψ0(x) = (2α)3/2 cosαx sin 2αx+ 2α(x− z) , ψ1(x) = 2α(x− z) sinαx+ cosαx√ 2α [sin 2αx+ 2α(x− z)] , ψ1(x) = i 2 √ 2α [ e−iαx − eiαx ] +O ( 1 x ) , x→ ±∞, (3.2) such that hψ0 = α2ψ0, ( h− α2 ) ψ1 = ψ0. The exceptional point E = α2 is a pole for the Green function G(x, x′;E) = [ πi k ψn(x>; k)ψn(x<;−k) ] ∣∣∣ k= √ E , x> = max{x, x′}, x< = {x, x′}, Im √ E > 0, (h− E)G = δ(x− x′). This pole is a pole of second order, it is replicated on both sides of the cut E > 0 and there are no other poles for G(x, x′;E). One can show [2] that the eigenfunctions and the associated function of h obey the biorthogo- nality relations,∫ +∞ −∞ ψ2 0(x) dx = 0, ∫ +∞ −∞ ψ0(x) [( k2 − α2 ) ψ(x; k) ] dx = 0, (3.3)∫ +∞ −∞ ψ0(x)ψ1(x) dx = 0, ∫ +∞ −∞ ψ1(x) [( k2 − α2 ) ψ(x; k) ] dx = 0, (3.4)∫ +∞ −∞ [(k2 − α2)ψ(x; k)] [( (k′)2 − α2 ) ψ(x;−k′) ] dx = ( (k′)2 − α2 )2 δ(k − k′), (3.5) where (3.3) are included in (3.5) due to the equality ψ0(x) = ∓i √ π α lim k→±α [( k2 − α2 ) ψ(x; k) ] (3.6) and (3.4) follow from (3.5) in view of (3.6) and of the equality ψ1(x) = ±i √ π α lim k→∓α { 1 2k ∂ ∂k [( k2 − α2 )] ψ(x; k) } − 1∓ 2iαz 4α2 ψ0(x). The resolution of identity constructed from ψ(x; k) holds [13], δ(x− x′) = ∫ L ψ(x; k)ψ(x′;−k) dk, (3.7) 1There is a misprint in the normalization of ψ0(x) and ψ1(x) in [2]. 14 A.A. Andrianov and A.V. Sokolov where L is an integration path in complex k plane, obtained from the real axis by its simultaneous deformation near the points k = −α and k = α upwards or downwards (the direction of this deformation is of no difference since for the points k = −α and k = α the sum of residues of the integrand is equal to zero). The direction of L is specified from −∞ to +∞. This resolution of identity is valid for test functions belonging to CLγ ≡ C∞R ∩ L2(R; (1 + \|x\|)γ), γ > −1 as well as for some bounded and even slowly increasing test functions and, in particular, for eigenfunctions ψ(x; k) and for the associated function ψ1(x). One can rearrange [13] the resolution of identity (3.7) for any ε ∈ (0, α) to the form δ(x− x′) = (∫ −α−ε −∞ + ∫ α−ε −α+ε + ∫ +∞ α+ε ) ψ(x; k)ψ(x′;−k) dk + 2 π cosα(x− x′)sin ε(x− x ′) x− x′ − 1 πα ψ0(x)ψ0(x ′) [ 1 ε [ 1− 2 sin2 ε 2 (x− x′) ] − ε 4α2 − ε2 cos 2α(x− x′) cos ε(x− x′)− 2α 4α2 − ε2 sin 2α(x− x′) sin ε(x− x′) ] − 1 π [ψ0(x)ψ1(x ′) + ψ1(x)ψ0(x ′)] ∫ 2α+ε 2α−ε cos t(x− x′)dt t (3.8) and, consequently, to the form δ(x− x′) = lim ε↓0 ′ {(∫ −α−ε −∞ + ∫ α−ε −α+ε + ∫ +∞ α+ε ) ψ(x; k)ψ(x′;−k) dk + 2 π cosα(x− x′)sin ε(x− x ′) x− x′ − 1 πα ψ0(x)ψ0(x ′) [ 1 ε [ 1− 2 sin2 ε 2 (x− x′) ] − ε 4α2− ε2 cos 2α(x− x′) cos ε(x− x′)− 2α 4α2− ε2 sin 2α(x− x′) sin ε(x− x′) ] − 1 π [ψ0(x)ψ1(x ′) + ψ1(x)ψ0(x ′)] ∫ 2α+ε 2α−ε cos t(x− x′)dt t } , (3.9) where the prime ′ at the limit symbol emphasizes that this limit is regarded as a limit in the space of distributions. One can reduce [13] the resolutions of identity (3.8) and (3.9) for test functions from CLγ , γ > −1 to the form δ(x− x′) = lim ε↓0 ′ {(∫ −α−ε −∞ + ∫ α−ε −α+ε + ∫ +∞ α+ε ) ψ(x; k)ψ(x′;−k) dk − 1 πεα [ 1− 2 sin2 ε 2 (x− x′) ] ψ0(x)ψ0(x ′) } (3.10) and for test functions from CLγ , γ > 1 to a more simple form δ(x− x′) = lim ε↓0 ′ {(∫ −α−ε −∞ + ∫ α−ε −α+ε + ∫ +∞ α+ε ) ψ(x; k)ψ(x′;−k) dk − 1 πεα ψ0(x)ψ0(x ′) } .(3.11) The latter of these resolutions seems to have a more natural form than the previous ones, but it cannot reproduce the normalizable eigenfunction ψ0(x) 6∈ CLγ ≡ C∞R ∩ L2(R; (1 + \|x\|)γ), γ > 1 Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 15 because of the biorthogonality relations. With the help of (3.2), Lemma 4.8 from [13] and the Jordan lemma one can check that lim ε↓0 ∫ +∞ −∞ [ 2 πεα sin2 ε 2 (x− x′)ψ0(x)ψ0(x ′) ] ψ0(x) dx = lim ε↓0 { ψ0(x ′) ∫ +∞ −∞ [ 2 πεα sin2 ε 2 (x− x′) ] × [ 2α cos2 αx (x− z)2 − sin 2αx[sin 2αx+ 4α(x− z)] 4α2(x− z)2 ψ2 0(x) ] dx } = lim ε↓0 { ψ0(x ′) ∫ +∞ −∞ [ 2 πεα sin2 ε 2 (x− x′) ] [ 2α cos2 αx (x− z)2 ] dx } (3.12) = lim ε↓0 {[ e±iε(z−x ′) − 4 α ε sin2 ε 2 (z − x′)± ie±2iαz sin ε(z − x′) ] ψ0(x ′) } = ψ0(x ′), where the upper (lower) signs correspond to the case Im z > 0 (Im z < 0). Hence, just the term 2 πεα sin2 ε 2 (x− x′)ψ0(x)ψ0(x ′) (3.13) in the resolutions of identity (3.8)–(3.10) gives an opportunity to reproduce ψ0(x) by these resolutions (cf. with the analogous results in Section 6.1 of [2], in Section 2 of [29] and in Section 2.4 of the present paper). It is interesting that the contribution of the term (3.13) in the resolutions of identity (3.8)–(3.10) is a singular discontinuous functional (see Remark 4.2 in [13]) which support consists of the only element – the infinity (cf. with the analogous comments in Section 2 of [29] and in Section 2.4 of the present paper). There is another way to transform the resolution (3.7) as well. This way was used for obtaining of the resolutions of identity (26) in [29] and (2.28) in Section 2.3. The integral in the right-hand part of (3.7) is understood as follows,∫ L ψ(x; k)ψ(x′;−k) dk = lim′ A→+∞ ∫ L(A) ψ(x; k)ψ(x′;−k) dk, (3.14) where L(A) is an integration path in complex k plane, obtained from the segment [−A,A] by its simultaneous deformation near the points k = −α and k = α upwards or downwards (the direction of this deformation is of no difference as well as in the case with L) and the direction of L(A) is specified from −A to A. Using (1) the fact that the integral in the right-hand part of (3.14) is a standard integral (not a distribution); (2) the fact that (k∓α)2ψ(x; k)ψ(x′;−k) is a holomorphic function of k in a neighborhood of k = ±α (see (3.1)); (3) the Leibniz formula and the formulae (3.2); (4) the notation L(k0; ε) with fixed k0 ∈ R and ε > 0 for the path in complex k plane defined by k = k0 + ε[cos(π − ϑ)± i sin(π − ϑ)], 0 6 ϑ 6 π, where the upper (lower) sign corresponds to the case of upper (lower) deformations in L and the direction of L(k0; ε) is specified from ϑ = 0 to ϑ = π; 16 A.A. Andrianov and A.V. Sokolov we can transform (3.14) as follows,∫ L ψ(x; k)ψ(x′;−k) dk = lim′ A→+∞ lim ε↓0 {(∫ −α−ε −A + ∫ α−ε −α+ε + ∫ A α+ε ) ψ(x; k)ψ(x′;−k) dk + 1∑ j=0 1 j! ∂j ∂kj [ (k + α)2ψ(x; k)ψn(x ′;−k) ]∣∣∣ k=−α ∫ L(−α;ε) dk (k + α)2−j + 1∑ j=0 1 j! ∂j ∂kj [ (k − α)2ψ(x; k)ψn(x′;−k) ]∣∣∣ k=α ∫ L(α;ε) dk (k − α)2−j } = lim′ A→+∞ lim ε↓0 {(∫ −α−ε −A + ∫ α−ε −α+ε + ∫ A α+ε ) ψ(x; k)ψ(x′;−k) dk + 1 4πα ψ0(x)ψ0(x ′) [∫ L(−α;ε) dk (k + α)2 + ∫ L(α;ε) dk (k − α)2 ] + 1 2π [ψ0(x)ψ1(x ′) + ψ1(x)ψ0(x ′)] [∫ L(α;ε) dk k − α − ∫ L(−α;ε) dk k + α ]} = lim′ A→+∞ lim ε↓0 {(∫ −α−ε −A + ∫ α−ε −α+ε + ∫ A α+ε ) ψ(x; k)ψ(x′;−k) dk − 1 πεα ψ0(x)ψ0(x ′) } = lim∗ ε↓0 {(∫ −α−ε −∞ + ∫ α−ε −α+ε + ∫ +∞ α+ε ) ψ(x; k)ψ(x′;−k) dk − 1 πεα ψ0(x)ψ0(x ′) } , (3.15) where the latter equality is considered as a definition for lim∗ and the limit limε↓0 is regarded as a pointwise one (not as a limit in a function space). The resolution of identity (3.15) is equivalent to (3.7), i.e. it is valid for all test functions for which (3.7) is valid (cf. with (3.11) and with the similar results in Section 2 of [29] and in Section 2.3 of the present paper). Let us note that the associated function ψ1(x) does not appear in the derived resolutions of identity and is not expandable with the help of the resolution (3.11). Thereby, this associated function does not belong to the physical state space (rigged Hilbert space). Let us notice also that the number (equal to 1) of linearly independent eigen- and (formal) associated functions of the Hamiltonian h for the eigenvalue E = α2 included into the resolutions of identity (3.11) and (3.15) is less than the order (equal to 2) of the pole E = α2 of the Green function G(x, x′;E). 4 Conclusions: indexes of exceptional points and SUSY We remark that, in general, one can introduce, at least, three different number indexes of exceptional point E = λ0 of a Hamiltonian h: (1) n1(λ0) to be a maximal number of linearly independent normalizable eigenfunctions and associated functions of h for the eigenvalue E = λ0; (2) n2(λ0) to be a maximal number of linearly independent eigenfunctions and formal asso- ciated functions of h for the eigenvalue E = λ0 appeared in the resolution of identity constructed from biorthogonal set of eigenfunctions and associated functions of h; (3) n3(λ0) to be an order of the pole in the point E = λ0 for the Green function for h as a function of E (in the case with the Hamiltonian h = hn (see Section 2), where the exceptional point E = λ0 = 0 coincides with the branch point of the Green function; it is natural to assume (see Section 2.3) that n3(0) = n). Resolutions of Identity for Some Non-Hermitian Hamiltonians. I 17 It can be proven (by methods of [7], see as well the example in Section 5.1 of [2]) that for an exceptional point outside of continuous spectrum all these indexes are identical, n1(λ0) = n2(λ0) = n3(λ0) and represent the algebraic multiplicity of the eigenvalue λ0. In the cases where an exceptional point is situated on the border of continuous spectrum or inside of it these indexes may be different. For example, in the case of an exceptional point E = 0 at the bottom of continuous spectrum of the Hamiltonian hn in Section 2, n1(0) = [ n+ 1 2 ] , n2(0) = n, n3(0) = n ⇒ n1(0) 6 n2(0) = n3(0) and in the case of an exceptional point E = α2 inside of continuous spectrum of the Hamiltonian h in Section 3, n1 ( α2 ) = 1, n2 ( α2 ) = 1, n3 ( α2 ) = 2 ⇒ n1 ( α2 ) = n2 ( α2 ) < n3 ( α2 ) . Thus, one can consider these indexes as different generalizations of the notion of algebraic multiplicity. One can use SUSY technique [14, 15, 16, 17, 18, 19, 20] in order to regulate the algebraic multiplicity (in any sense mentioned above) of an exceptional point in the spectrum of a SUSY partner Hamiltonian with respect to the order of this exceptional point in the spectrum of a given Hamiltonian (originally proposed in [21] and elaborated in details in [3, 4]): (1) in order to increase the multiplicity of an exceptional point λ0 in the spectrum of a SUSY partner Hamiltonian with respect to its order in the spectrum of a given Hamiltonian, one must take a formal eigenfunction (and a chain of formal associated functions) for the spectral value λ0 of the latter Hamiltonian as transformation function(s) which tends (tend) to infinity for x→ ±∞; (2) in order to decrease the multiplicity of an exceptional point λ0 in the spectrum of a SUSY partner Hamiltonian with respect to its multiplicity in the spectrum of a given Hamilto- nian, one must take a normalizable eigenfunction (and a chain of normalizable associated functions) of the latter Hamiltonian for eigenvalue λ0 as transformation function(s). These statements can be clarified by the following simple example. For the eigenvalue E = λ0 = 0 of the Hamiltonian hn, n = 1, 2, 3, . . . in Section 2 there is a chain (2.1) of the eigenfunction ψn0(x) and associated functions ψnl(x), l = 1, . . . , [(n− 1)/2]. As well one can check that for the spectral value E = 0 of the Hamiltonian hn, n = 0, 1, 2, . . . there is a chain of the formal eigenfunction ϕn0(x) and formal associated functions ϕnl(x), ϕn0(x) = (x− z)n+1, ϕnl(x) = (−1)l(2n+ 1)!! (2l)!!(2n+ 2l + 1)!! (x− z)n+2l+1, hnϕn0 = 0, hnϕnl = ϕn,l−1, l = 1, 2, 3, . . . , which tend to infinity for x → ±∞. Thus, it can be easily found that a) if to use ϕnl(x), l = 0, . . . ,m as transformation functions for hn, then the resulting Hamiltonian is hn+m+1 with the exceptional point E = 0 of larger algebraic multiplicity (except for the case with the indexes n1(0), m = 0 and odd n, where the indexes n1(0) for hn and hn+m+1 ≡ hn+1 are equal), and b) if to use ψnl(x), l = 0, . . . ,m 6 [(n − 1)/2] as transformation functions for hn, then the resulting Hamiltonian is hn−m−1 with the exceptional point E = 0 of smaller algebraic multiplicity (except for the case with indexes n1(0), m = 0 and even n, where the indexes n1(0) for hn and hn−m−1 ≡ hn−1 are equal). 18 A.A. Andrianov and A.V. Sokolov Acknowledgments This work was supported by Grant RFBR 09-01-00145-a and by the SPbSU project 11.0.64.2010. The work of A.A. was also supported by grants 2009SGR502, FPA2007-66665 and by the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042). 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Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum

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