Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.


Introduction
This part is a continuation of the Part I [1] where resolutions of identity for certain non-Hermitian Hamiltonians were constructed of biorthogonal sets of their eigen-and associated functions. The spectral problem was defined on entire axis. Non-Hermitian Hamiltonians were taken with continuous spectrum and they were endowed with an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum or an exceptional point situated inside of continuous spectrum. In the present work (Part II) the detailed rigorous proofs are given for the resolutions of identity in both cases. Moreover the reductions of the derived resolutions of identity under narrowing of the classes of employed test functions in the Gel'fand triple [2] are built. In Section 2 the definitions of the employed spaces of test functions and distributions are given. In Section 3 the proofs of the initial resolution of identity and of its reduced forms for restricted spaces of test functions are elaborated for an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum. In Section 4 the analogous proofs of resolutions of identity are presented for an exceptional point situated inside of continuous spectrum. if lim n→+∞ +∞ −∞ |ϕ n (x) − ϕ(x)| 2 (1 + |x|) γ dx = 0, and for any x 1 , x 2 ∈ R, x 1 < x 2 and any l = 0, 1, 2, . . . , We shall denote the value of a functional f on ϕ ∈ CL γ conventionally as (f, ϕ). A linear functional f is called continuous if for any sequence ϕ n ∈ CL γ , n = 1, 2, 3, . . . convergent in CL γ to zero the equality lim n→+∞ (f, ϕ n ) = 0 is valid. The space of distributions over CL γ , i.e. of linear continuous functionals over CL γ , is denoted CL ′ γ . The sequence f n ∈ CL ′ γ , n = 1, 2, 3, . . . is called convergent in CL ′ γ to f ∈ CL ′ γ , if for any ϕ ∈ CL γ the relation takes place, lim n→+∞ (f n , ϕ) = (f, ϕ).
A functional f ∈ CL ′ γ is called regular if there is f (x) ∈ L 2 (R; (1 + |x|) −γ ) such that for any ϕ ∈ CL γ the equality holds. In this case we shall identify the distribution f ∈ CL ′ γ with the function f (x) ∈ L 2 (R; (1+ |x|) −γ ). In virtue of the Bunyakovskii inequality, it is evident that L 2 (R; (1 + |x|) −γ ) ⊂ CL ′ γ and this inclusion is continuous. For any γ 1 < γ 2 there is a continuous inclusion CL γ 2 ⊂ CL γ 1 . Let us also notice that the Dirac delta function δ(x − x ′ ) belongs to CL ′ γ for any γ ∈ R.
3 Proofs of resolutions of identity for the model Hamiltonians with exceptional point of arbitrary multiplicity at the bottom of continuous spectrum 3.1 Proof of the biorthogonality relations between eigenfunctions for continuous spectrum Let us start proofs by proving of the biorthogonality relation between eigenfunctions ψ n (x; k) for the continuous spectrum of the Hamiltonian h n , n = 0, 1, 2, . . . (see (2.17) of Part I). Proof of this biorthogonality relation (3.1) is based on the following Lemmas 3.1-3.3.
Lemma 3.1. Suppose that the functions ψ n (x; k), n = 0, 1, 2, . . . are defined by the formula (2.6) of Part I for any x ∈ R, k ∈ C, k = 0 and fixed z ∈ C, Im z = 0. Then for any n = 1, 2, 3, . . . , R > 0, k ∈ C and k ′ ∈ C the following relation holds, Proof . Let us check first that This equality can be derived with the help of (2.3), (2.6) and (2.9) of Part I and integration by parts, The equality (3.2) follows from (3.3) by induction, in view of the relation Lemma 3.1 is proved.

Proofs of the resolutions of identity
Proof of the initial resolution of identity (2.18) of Part I is based on the following Lemmas 3.4-3.6.

Lemma 3.4. Suppose that
(1) the functions ψ n (x; k), n = 0, 1, 2, . . . are defined by the formula (2.6) of Part I for any x ∈ R, k ∈ C, k = 0 and fixed z ∈ C, Im z = 0; Then for any n = 1, 2, 3, . . . , x ∈ R and x ′ ∈ R the following relation holds, Proof . Let us check first that This equality can be derived with the help of (2.3), (2.6) and (2.9) of Part I and of integration by parts: The equality (3.5) follows from (3.6) by induction in view of the relation Lemma 3.4 is proved.
Lemma 3.5. For any x ′ ∈ R, z ∈ C, Im z = 0, l = 0, 1, 2, . . . , m = 1, 2, 3, . . . and γ > 1 − 2m the following relation takes place, Proof . It is true that Thus, in view of the Riemann theorem, in order to prove the lemma, it is sufficient to prove that for any ϕ(x) ∈ CL γ the fraction ϕ(x)/(x − z) m belongs to L 1 R . The latter is valid by virtue of the Bunyakovskii inequality: where the condition γ > 1 − 2m is taken into account. Lemma 3.5 is proved.
In the conditions of Lemma 3.4, in view of (2.6) from Part I by virtue of Lemma 3.5 for any n = 1, 2, 3, . . . , l = 0, . . . , n − 1 and γ > −2l − 1, the following relation holds, Lemma 3.6. For any x ′ ∈ R, z ∈ C, Im z = 0, n = 0, 1, 2, . . . and γ > −2n − 1 the following relation is valid, Proof . It is true that Thus, to prove the lemma, it is sufficient to prove that for any ϕ(x) ∈ CL γ , γ > −2n − 1 the equality takes place. For this purpose let us consider the function By virtue of the Bunyakovskii inequality for arbitrary δ > 0, and, moreover, it is evident that Hence, by virtue of the Riemann theorem, Thus, Lemma 3.6 is proved.
Validity of the resolution of identity (2.18) of Part I in CL ′ γ for any γ > −1 is a corollary of the following theorem.

Theorem 3.2. Suppose that
(1) the functions ψ n (x; k), n = 0, 1, 2, . . . are defined by the formula (2.6) of Part I for any x ∈ R, k ∈ C, k = 0 and fixed z ∈ C, Im z = 0; Then for any γ > −1, x ′ ∈ R and n = 0, 1, 2, . . . the following relation holds, The statement of Theorem 3.2 follows from Lemmas 3.4 and 3.6 and from Corollary 3.2. The applicability of the resolution of identity (2.18) of Part I for some bounded and slowly increasing test functions is based on the next theorem.
Then for any κ ∈ [0, 1), k 0 ∈ R, x ′ ∈ R and n = 0, 1, 2, . . . the following relation is valid, Proof . In the case n = 0 in view of (3.7) the proof can be easily realized in the same way as for Theorem 2 from Appendix B of [4]. Thus, we present the proof for the case n = 1, 2, 3, . . . with upper signs in (3.8) only, taking into account that the proof for the case with lower signs is quite similar. In order to prove Theorem 3.3 in this case we employ Lemmas 3.4 and 3.6, Corollary 3.2 and the fact that Then it is sufficient to prove that In turn, to prove the latter, in view of (2.6) from Part I, (3.9) and Lemma 3.5, it is sufficient to prove that The equality (3.10) follows from the Riemann theorem and the chain of transformations, derived with help of integration by parts. Thus, Theorem 3.3 is proved.
Proof of Lemma 3.7 is analogous to the proof of Lemma 2 from Appendix B of [4]. The resolution of identity (2.37) of Part I is a corollary of the resolution of identity (2.36) of Part I and of the following Lemma 3.8. Lemma 3.8. For any γ > 1, x ′ ∈ R and z ∈ C, Im z = 0 the relation takes place, Proof of Lemma 3.8 is analogous to the proof of Lemma 3 from Appendix of [3]. The resolution of identity (2.38) of Part I is a corollary of the resolution of identity (2.37) of Part I and of the following Lemmas 3.9 and 3.10. Lemma 3.9. For any γ > 3, x ′ ∈ R and z ∈ C, Im z = 0 the relation is valid, Proof . It is true that Thus, to prove the lemma it is sufficient to establish that for any ϕ(x) ∈ CL γ , γ > 3, the relation is valid. But its validity follows from the chain of inequalities, derived with the help of the Bunyakovskii inequality. Lemma 3.9 is proved.
Lemma 3.10. For any γ > 3, x ′ ∈ R and z ∈ C, Im z = 0 the relation takes place, Proof . It is true that Thus, to prove the lemma it is sufficient to establish that for any ϕ(x) ∈ CL γ , γ > 3, the relation is valid. But its validity follows from the chain of inequalities, derived with the help of the Bunyakovskii inequality. Lemma 3.10 is proved.
Remark 3.4. Let us consider the functionals each of which is defined by a related expression in the set for all test functions ϕ(x) ∈ CL γ , γ ∈ R, for which the limit from (3.12) corresponding to (3.11) exists. It follows from Lemmas 3.9 and 3.10 that these functionals are trivial (equal to zero) for any γ > 3, but at the same time in view of the formulae (2.39) and (2.40) from [1] these functionals are nontrivial (different from zero) for any γ < 3. By virtue of Lemmas 3.9 and 3.10 the restrictions of the functionals (3.11) on the standard space D(R) ⊂ CL γ , γ ∈ R are equal to zero. Hence, the supports of these functionals for any γ ∈ R do not contain any finite real number. On the other hand, one can represent a test function ϕ(x) ∈ CL γ , γ ∈ R for any R > 0 as a sum of two functions from CL γ in the form for any x ∈ [0, 1] and η(x) ≡ 0 for any x > 1. In view of Lemmas 3.9 and 3.10 the values of the functionals (3.11) for ϕ(x) are equal to their values for the second term of (3.13) for any arbitrarily large R > 0. Hence, the values of the functionals (3.11) for a test function depend only on the behavior of this function in any arbitrarily close (in the conformal sense) vicinity of the infinity and are independent of values of the function in any finite interval of real axis. In this sense the supports of the functionals (3.11) for any γ < 3 consist of the unique element which is the infinity. At last, since (i) for any ϕ(x) ∈ CL γ and γ ∈ R the relation holds; (ii) the restrictions of the functionals (3.11) on D(R) are zero for any γ ∈ R and (iii) the functionals (3.11) are nontrivial for any γ < 3, so the latter functionals are discontinuous for any γ < 3.

Proofs of resolutions of identity for the model Hamiltonian with exceptional point inside of continuous spectrum
Proof of the initial resolution of identity (3.7) of Part I is based on the following Lemmas 4.1-4.3.

Lemma 4.1. Suppose that
(1) the functions ψ(x; k), ψ 0 (x) and ψ 1 (x) are defined by the formulas (3.1) and (3.2) of Part I for fixed α > 0, z ∈ C, Im z = 0 and any x ∈ R, k ∈ C, k = ±α; (2) L(A) is an integration path in complex k plane, obtained from the segment [−A, A], A > α by its simultaneous deformation near the points k = −α and k = α upwards or downwards and the direction of L(A) is specified from −A to A.
Then for any x, x ′ ∈ R and A > α the following relation is valid, Proof . With the help of (3.1) and (3.2) of Part I and certain identical transformations one can rearrange the left-hand part of (4.1) to the form, where from the equality (4.1) follows trivially. Lemma 4.1 is proved.
In the conditions of Lemma 4.1 for any x ′ ∈ R and γ > −1 the following relation holds, Proof of Lemma 4.2 in view of (3.2) of Part I is quite similar to the proof of a more complicated Lemma 3.2 from Section 3.1.
Therefrom it follows that where D is a finite constant by virtue of (3.2) of Part I. The statement of Lemma 4.3 is valid in view of the following chain of inequalities obtained with the help of (4.2) and the Bunyakovskii inequality, where ϕ(x) is any function from CL γ , γ > −1. Lemma 4.3 is proved.
Validity of the resolution of identity (3.7) of Part I in CL ′ γ for any γ > −1 is a corollary of the following theorem. (1) the function ψ(x; k) is defined by the formula (3.1) of Part I for fixed α > 0, z ∈ C, Im z = 0 and any x ∈ R, k ∈ C, k = ±α; (2) L(A) is an integration path in complex k plane, obtained from the segment [−A, A], A > α by its simultaneous deformation near the points k = −α and k = α upwards or downwards and the direction of L(A) is specified from −A to A.
Then for any γ > −1 and x ′ ∈ R the following relation holds, Then for any κ ∈ [0, 1), k 0 ∈ R and x ′ ∈ R the following relation holds, Proof of Theorem 4.2 is quite analogous to the proof of Theorem 2 from Appendix B of [4] and it is based on the inequalities from Lemma 4.4. of Part I for test functions which are linear combinations of functions η(±x)e ik 0 x |x| κ , in general, with different κ ∈ [0, 1) and k 0 ∈ R and functions from CL γ , in general, with different γ > −1. In particular, these theorems guarantee applicability of (3.7) of Part I to the eigenfunctions ψ(x; k) and to the associated function ψ 1 (x) of the Hamiltonian h (see Part I).
The resolutions of identity (3.8) and (3.9) of Part I are corollaries of the resolution of identity (3.7) of Part I and of the following Lemma 4.5.
Then for any x, x ′ ∈ R and ε ∈ (0, α) the following relation is valid, (4.7) (4.7) follows trivially from the same representation of the integrand ψ(x; k)ψ(x ′ ; −k) as in the proof of Lemma 4.1.
Proof of the resolution of identity (3.10) of Part I is based on the following Lemmas 4.6 and 4.7.
Lemma 4.6. In the conditions of Lemma 4.5 for any x ′ ∈ R and γ > −1 the following relation takes place, lim ′ γ ε↓0 ψ 0 (x) ε cos 2α(x − x ′ ) cos ε(x − x ′ ) + 2α sin 2α(x − x ′ ) sin ε(x − x ′ ) = 0. where ψ 0 (x) is the eigenfunction (3.2) of Part I, which is defined by the expression for all test functions ϕ(x) ∈ CL γ , γ ∈ R, for which the limit (4.10) exists. It follows from Lemma 4.8 that the functional (4.9) is trivial (equal to zero) for any γ > 1, but at the same time, in view of the formula (3.12) from [1], this functional is nontrivial (different from zero) for any γ < 1. By virtue of Lemma 4.8 the restriction of the functional (4.9) on the standard space D(R) ⊂ CL γ , γ ∈ R is equal to zero. Hence, the support of this functional for any γ ∈ R does not contain any finite real number. On the other hand, one can represent any test function ϕ(x) ∈ CL γ , γ ∈ R for any R > 0 as a sum of two functions from CL γ in the form where η(x) ∈ C ∞ R , η(x) ≡ 1 for any x < 0, η(x) ∈ [0, 1] for any x ∈ [0, 1] and η(x) ≡ 0 for any x > 1. In view of Lemma 4.8 the value of the functional (4.9) for ϕ(x) is equal to its value for the second term of (4.11) for any arbitrarily large R > 0. Hence, the value of the functional (4.9) for a test function depends only on the behavior of this function in any arbitrarily close (in the conformal sense) vicinity of the infinity and is independent of values of the function in any finite interval of real axis. In this sense the support of the functional (4.9) for any γ < 1 consists of the unique element which is the infinity. At last, since (i) for any ϕ(x) ∈ CL γ and γ ∈ R the relation lim γ R→+∞ η(|x| − R)ϕ(x) = ϕ(x) holds; (ii) the restriction of the functional (4.9) on D(R) is zero for any γ ∈ R and (iii) the functional (4.9) is nontrivial for any γ < 1, so the functional (4.9) for any γ < 1 is discontinuous.