Two-term self-adjoint fourth-order differential operator with summable potential on the unit interval is considered. High energy eigenvalue asymptotics and the trace formula for this operator are obtained.
Consider self-adjoint fourth-order operators acting in the Hilbert space and defined by
where q is real potential and . The operator is determined on the domain
A great number of papers are devoted to the eigenvalue asymptotics for second-order operators. We mention only the books of Marchenko [17] and Levitan and Sargsyan [16], the review Fulton and Pruess [12], and see also the references therein.
The spectral properties of fourth-order differential operators with various boundary conditions are studied in many papers. Firstly, we discuss the asymptotic properties of general fourth-order differential operators on the unit interval. Caudill, Perry, and Schueller [10] described iso-spectral potentials for these operators. McLaughlin [18] investigated inverse spectral problems for fourth-order operators with the Neumann type boundary conditions. Polyakov [25] obtained the eigenvalue asymptotics and the estimates of spectral projections for these operators. Badanin and Korotyaev determined eigenvalue asymptotics and trace formulas for self-adjoint fourth-order operators on the circle [5] and on the unit interval [7]. The fourth-order operators with periodic coefficients on the line were considered by Badanin and Korotyaev [3,6]. Moreover, these authors [2] are studied the periodic operator H. Gunes, Kerimov, and Kaya [15] obtained the eigenvalue asymptotics for general fourth-order differential operators with periodic (antiperiodic) boundary conditions and proved that the system of eigenfunctions and associated functions of this operator form a basis in the space , . Also these operators were considered by Polyakov [26,28]. The spectral properties of the periodic Euler-Bernoulli equation were studied by Papanicolaou [22,24].
Spectral asymptotics for higher-order operators are much less investigated. Numerous results about the regular boundary value problems for these operators are expounded by Naimark [20]. High energy asymptotics of the eigenvalues for even-order operator are determined by Akhmerova [1], Badanin and Korotyaev [4], Mikhailets and Molyboga [19], Polyakov [27].
Secondly, we describe results about trace formulas for fourth and higher-order operators. We mention only several results that are closely related to our ones. Sadovnichii and Podolskii [30] represented the survey about trace formulas. Shevchenko [31] determined the trace formula of two-term arbitrary order differential operators for general Birkhoff regular boundary conditions with smooth potentials. The trace formulas for the operator H were obtained by Sadovnichii [29] (under the Dirichlet type boundary conditions) and Bayramov, Oer, Öztürk Uslu, and Kizilbudak Caliskan [9] (under the periodic boundary conditions). Papanicolaou [23] deduced the trace formula for higher-order Schrödinger operators with Neumann or Robin boundary conditions. Nazarov, Stolyarov and Zatitskiy [21] determined the first-order trace formula for a regular differential operators on the segment perturbated by a multiplication operator. Gül [13] proved the first regularized trace formula for the self-adjoint operator H with bounded operator coefficient. Gül and Ceyhan [14] obtained the second regularized trace formula for this operator.
The main goal of the present paper is to determine eigenvalue asymptotics and the trace formula for the operator H.
The operator H has pure discrete spectrum (see [20, Ch. I]). Introduce the fundamental solutions , , of equation
satisfying the conditions , j, , where is the Kronecker symbol. Each , , is an entire function.
The spectrum consists of real or non-real eigenvalues and satisfies , where D is an entire function given by
Consider the unperturbed operator under the boundary conditions (1.1). In this case the function D has the form
see also (2.12). Here and below we use the spectral parameter
The eigenvalues of are the zeros of and have the form , . These eigenvalues are simple.
We denote the eigenvalues of the perturbed problem by , , (see details in Section 3.1). Our first main results are devoted to the high energy asymptotics of these eigenvalues in the case . We give these asymptotics in terms of Fourier coefficients of q. In order to determine the trace formula for the operator H we need an additional condition . We also derive the eigenvalue asymptotics in this case.
Introduce the Fourier coefficients
Now we can formulate the main results of our paper.
Letand letbe large enough. Then the eigenvaluesare real and simple. Moreover, they satisfyIf, in addition,, then
The proof of Theorem 1 is based on the matrix form of the Birkhoff method. This method was used in [6] and [8].
We use asymptotics (1.5) in order to obtain the trace formula for the operator H (see Theorem 2 below).
Our second results are devoted to the trace formula for the operator H. Consider the shifted operator , given by (1.1), where , , and . We denote by the eigenvalues of the operator .
Let. Then there existssuch that each function,,, belongs to the space. Moreover, the following trace formula holds:the series converges absolutely and uniformly on.
(i) The asymptotics (1.5) show that if , then the series (1.6) converges absolutely and uniformly on . The proof of other results is based on the asymptotic analysis of difference of resolvents of the perturbed operator H and the unperturbed operator .
(ii) Nazarov, Stolyarov and Zatitskiy [21] determined “formal” the first-order trace formula for a regular differential operators, but they did not obtain conditions under which these series converge. Our result coincides with this trace formula, but we prove that corresponding series are convergence.
(iii) Papanicolaou [23] obtained the trace formula for higher-order Schrödinger operators with Neumann or Robin boundary conditions. Our result also coincides with this trace formula, but we use another method for the proof.
Fundamental solutions
Fundamental matrix
It will be convenient for us, instead of the fundamental solutions , , introduced in §1, to use other fundamental solutions , whose asymptotics are well controlled. In order to describe these fundamental solutions, we introduce additional notation.
Recall that , , , where
If , then , where
Introduce the numbers , . Then the following estimates hold:
Note that unperturbed equation (1.2) with has the fundamental solutions
Consider the perturbed equation (1.2). Let be large enough and let , where
Then equation (1.2) has the fundamental solutions , , , satisfying the asymptotics
as , , uniformly on any bounded subset of (see [20]).
Introduce the fundamental matrix , , of equation (1.2) by
The matrix-valued function A satisfies the equation
Rewrite the determinant D given by (1.3) in terms of fundamental solutions ϕ. Introduce the function
where the matrix-valued function ϕ has the form
The function ξ, like D, is the characteristic function of the spectrum of the operator H, but, unlike D, it is not an entire function of the variable λ. However, the function ξ has a well-controlled asymptotic behavior at high energy. The function ξ is analytic in and the function D given by (1.3) satisfies
The proof of this equation repeats the arguments from [8, Lemma 3.2]. The function D is entire, then the identity (2.6) can be extended analytically from onto .
Consider the unperturbed case . Let . In this case the fundamental matrix has the form
where
and , , are given by (2.1). Using the identity , we obtain
The matrix-valued function has the form
Then the function satisfies
where the entire function has the form
We analyse equation (1.2) by using the matrix form of the Birkhoff method for asymptotic analysis of higher-order equations (see [4,6,8]). In essence, the Birkhoff method is a peculiar inversion of the operator on the left-hand side of equation (2.3). This inversion is carried out in such a way that the result is an integral equation with a contracting kernel. This integral equation is solved by iterations. One problem is that the matrix coefficient on the right-hand side of equation (2.3) grows in the λ-variable. Therefore, we preliminarily transform equation (2.3) into equation (2.15) having a diagonal coefficient on the left-hand side and a decreasing matrix coefficient on the right-hand side.
Define the matrix-valued function by
In the unperturbed case , where is defined by (2.7), satisfies , where
We prove the following result.
Letand let, whereis large enough. Then the matrix-valued functiongiven by (
2.13
) satisfies the equationwhere the diagonal matrix T and the matrixhave the formwith
Substituting the definition (2.13) into equation (2.3) and using the identity
where is defined in (2.3), we obtain the following equation for the matrix-valued function :
Adding the term to both sides of this equation we obtain (2.15). □
Consider the smooth potential . In this case, we may reduce equation (2.15) to equation (2.22), where the matrix coefficient on the right-hand side decreases as . In order to obtain these results we use the method from [11, Ch. V.1.3]. We introduce a new unknown matrix-valued function by
where is the solution of equation (2.15).
We search the matrix-valued function U in the form
We choose the matrix U so that the coefficient on the right-hand side of equation (2.22) decreases as . It turns out that the matrix W must satisfies the following identity (see the proof of Lemma 2 below):
where . Direct calculations give
Thus, we obtain the following results.
Letand let, whereis large enough. Then the matrix-valued function,, given by (
2.18
), satisfies the equationwhere T is defined by (
2.16
) andasuniformly onand the matrix W given by (
2.21
).
Let . Substituting (2.18) into (2.15), we obtain
The identity (2.19) yields the asymptotics
uniformly on . Using (2.19) and (2.25), we get
uniformly on . These asymptotics give
uniformly on . Substituting (2.20) into (2.26), we obtain
where T and are defined by (2.16) and (2.23), respectively. Substituting the last identity into equation (2.24), we have (2.22). □
The Birkhoff method
The fundamental matrix contains both exponentially increasing, bounded, and decreasing entries at high energy. Therefore, asymptotic analysis of this matrix is rather difficult. In various combinations of entries of the fundamental matrix, the contribution of bounded and decreasing entries completely disappears against the background of the contribution of increasing ones. Birkhoff’s method gives a factorization of the fundamental matrix, where the exponential entries are separated into a separate diagonal matrix. This allows good control over their contribution when calculating the asymptotics.
Here we write out the basic formulas for the matrix form of the Birkhoff method (see [4,6,8]). Consider the differential equation
on the interval with the unknown -matrix-valued function , where , is large enough, , and the -matrix-valued functions and satisfy the following conditions:
is diagonal;
and for all ;
for a. e. the functions and are analytic in and
as , , uniformly in , where the matrix is given by (2.14) and the -matrix-valued function ;
F is off-diagonal, that is, , .
Note that equations (2.15) and (2.22) have the form (2.27). In fact, using the Birkhoff method we rewrite the differential equation (2.27) in the form of a specific Fredholm integral equation with a small kernel at high energy.
Let K be an integral operator in the space of matrix-valued functions given by
for all , where is large enough and
Then for the integral operator K is a contraction. Therefore, the matrix-valued integral equation
has a unique solution . Moreover, and each matrix-valued function , , is analytic on and satisfies the asymptotics
The integral equation (2.30) and the differential equation (2.27) are equivalent in the following sense.
Let,, whereis large enough. Then the matrix-valued functiongiven by the identitysatisfies the differential equation (
2.27
) if and only ifis a solution of the integral equation (
2.30
).
The proof repeats the arguments from [8, Theorem 4.5].
Representation of the fundamental matrix
Using (2.32), in the following lemma we obtain the factorization of the fundamental matrix A of equation (1.2). We represent the matrix A as a product of the bounded matrix , the simple matrix Λ, and the diagonal matrix . Thus, all exponentially increasing terms are removed from A into this diagonal matrix. In fact, we obtain two factorizations: the factorization (2.38) for nonsmooth potential and the factorization (2.41) for the smooth one. Despite the fact that the factorization (2.38) is also true in the case of smooth potential, it is inconvenient for calculating sharp eigenvalues asymptotics, since the remainder term in has the order . In the asymptotics (2.41), the remainder term in has the order , therefore, it is much more convenient for obtaining sharp eigenvalues asymptotics.
Introduce the matrix-valued function , , , by the formulas
where the matrix-valued functions , , defined by
where Q and W are defined by (2.17) and (2.21), respectively. Note that each matrix-valued function , , is analytic and bounded in .
Introduce the functions , , k, , , by
where W is defined by (2.21). Note that the functions , , are analytic and bounded in .
We prove the following result.
Letand letfor somelarge enough. Then
The fundamental matrix A of equation (
1.2
) satisfies the asymptotics
where Λ is given by (
2.8
), the diagonalmatrix-valued function T has the form (
2.16
), and the matrix-valued functiondefined by (
2.33
)–(
2.35
). Moreover, the fundamental solutions,,, given by (
2.2
) have the formandas,, for, uniformly on, where,, are defined by (
2.37
).
Let. Then
where T is defined by (
2.16
), U is given by (
2.19
), and the matrix-valued functiondefined by (
2.33
)–(
2.35
). Moreover, the fundamental solutions,,, have the form (
2.39
) and (
2.40
) for, where,, are defined by (
2.37
).
(i) It follows from (2.13) that , where is the solution of equation (2.15). Comparing (2.15) and (2.27), we see that , , , where T and are defined by (2.16). Using (2.32), we get
where is the solution of the integral equation (2.30) with , , . Substituting (2.29) into (2.31) and using the second identity in (2.16), we obtain
where is given by (2.33)–(2.35), as , , uniformly in . Substituting (2.43) into (2.42), we get (2.38). The identities , , , imply (2.39) and (2.40) for , where , , have the form (2.37).
(ii) It follows from (2.13) and (2.18) that , where is the solution of equation (2.22). Comparing (2.22) and (2.27), we obtain , , , where T and are defined by (2.16) and (2.23), respectively. Using (2.32) and (2.18), we get
where is the solution of the integral equation (2.30) with , , . The asymptotics (2.31) yields
as , , uniformly in . Substituting the asymptotics (2.23) into (2.29), we get
where l, , and is given by (2.33)–(2.35). Substituting these relations into (2.45), we have
Substituting this asymptotics into (2.44), we get (2.41). The equations (2.41) and (2.19) imply
where W has the form (2.21). The identities , , , imply (2.39) and (2.40) for , where , , are defined by (2.37). □
Counting lemma
Now we prove Counting Lemma for the zeros of the function D defined by (2.6). Introduce the domains , , and the contours by
Always below the asymptotics for are uniform with respect to in the corresponding sector.
Let. Then
The functionsatisfiesas,. Moreover,as,.
For each integerlarge enough the function D has (counting with multiplicities)zeros in the diskand for eachit has exactly one simple real zero in the domain.
(i) It follows from (2.38) that . The definition (2.8) yields . This gives (2.46).
Let and . Substituting (2.2) into (2.5) we obtain
Let, in addition, . Substituting (2.48) into (2.4) and using (2.11), we get
Substituting (2.46) and (2.49) into the identity (2.6), we obtain (2.47).
(ii) Let be integer and large enough and let be another integer. Let λ belong to the contours , , and for all . Using (2.47), we get
on all contours. Hence, by Rouche’s theorem, D has the same number of zeros as the function in each of the bounded domains and the remaining unbounded domain. The function has exactly one simple zero in , . Therefore, the function D has zeros in the disk and for each exactly one simple zero in the domain . Since can be chosen arbitrary large, the statement of Lemma follows.
We prove that the zero of D in the domain , , is real. The facts above shows that the function D has one zero μ in . If , then there is another zero . Therefore, there are two zeros of D in . We get a contradiction with the previous statement. Hence μ is real. □
Asymptotics of the determinant of fundamental matrix
Lemma 5 shows that the eigenvalues at high energy lie in the vicinities of the unperturbed eigenvalues. Our next goal is to obtain high energy asymptotics for the eigenvalues. For this we need some preliminary results.
Introduce the sector .
The matrix ϕ, given by (2.5), has the form (2.10) in the unperturbed case. This shows that the first column of this matrix contains exponentially decreasing terms as in , and the fourth column contains exponentially increasing ones. Using this fact, we can determine the main term in the asymptotics of the characteristic determinant ξ, given by (2.4). This term is expressed in terms of linear combinations for product of second-order determinants. Thus, we obtain the following results.
Letand let. Then,. Moreover,where
Let and . Then . Consider the identity (2.4). Direct calculations yield
The asymptotics (2.2) imply
and
The estimates and the asymptotics (2.2) give
and
Substituting these asymptotics into (2.53), we obtain (2.50). Moreover, using again the asymptotics (2.2) and the estimates , we get
This gives the statement of the lemma. □
Eigenvalues for
Rough eigenvalue asymptotics
Recall that the eigenvalues of the operator H are zeros of the entire function D given by (1.3).
We index the eigenvalues of the function D by the following way. It follows from Lemma 5 that the function D has (counting with multiplicities) zeros in the disk for each integer large enough. The eigenvalues inside this domain are labeled by
Each domain , , contains one real eigenvalue that we denote by .
We determine the rough asymptotics of the large eigenvalues . The identity (2.6) and the asymptotics (2.46) show that the large eigenvalues are zeros of the functions defined by (2.4). The function is analytic in , where is large enough.
The eigenvaluessatisfy
Let , . Lemma 5(ii) yields , where . Substituting the asymptotics (2.2) into (2.51) and (2.52), we obtain
and
Substitute these asymptotics into (2.50). Then
The equation gives . Therefore, . This yields (3.1). □
Sharp eigenvalue asymptotics
Let . Then the fundamental matrix A satisfies (2.38). In order to clarify the eigenvalue asymptotics we need to improve the asymptotics of the function ξ. Introduce the functions α and β by the formulas
where are elements of the matrix and T is defined by (2.16). The functions α and β are real-analytic functions on the domain . The asymptotics (2.28) and the definition (2.14) give , , , .
We prove the following results.
Letand let. Then the functions ξ defined by (
2.50
) have the asymptoticswhereand, and,,,, satisfy the asymptoticsHere,,, are defined by (
2.37
).
Let , . We prove the statement of this lemma in the case . The proof in the case is similar. The formula (2.37) gives
Substituting (2.39) and (2.40) into (2.51) and using (3.2), (3.5), we get
and
Recall that the functions , k, , are bounded (see §2.3). Substituting (2.39) and (2.40) into (2.52) and using again (3.2) and (3.5), we obtain
and
Substituting (3.6)–(3.9) into (2.50), we obtain (3.3). □
Now we determine sharp eigenvalues asymptotics for the case . First, we obtain asymptotics of the functions and defined by (3.4). Introduce the sequence
Note that , as .
Let . The first definition (3.4) and the first identity (2.37) yield
Substituting (2.33)–(2.35) into (3.12) and using the first identity in (2.36), we have
The identity gives the first asymptotics in (3.11).
Similarly, the second definition (3.4) and the first identity (2.37) imply
Substituting (2.33)–(2.35) into (3.13) and using the first identity in (2.36), we obtain
This yields the second asymptotics in (3.11). □
Second, we determine asymptotics of the eigenvalues .
Let. Then the eigenvaluessatisfy the asymptotics (
1.4
).
Let , . Lemma 7 shows , . Substituting the asymptotics (3.11) into (3.3), we get
where α, β, and have the form (3.2) and (3.10), respectively. Substituting the asymptotics
and
into (3.14) and using the asymptotics , we obtain
The equation gives
Then identity implies
which yields the asymptotics (1.4). □
The case
Now we determine sharp eigenvalues asymptotics for the case . First, we determine asymptotics of the functions and defined by (3.4). Introduce the sequence
Note that , as .
Let. Letand let. Thenwherehas the form (
4.1
) and
Let . The first definition (3.4) and the second formula (2.37) yield
The identity (2.21) gives
Substituting (2.33)–(2.35) into this representation and using the second identity in (2.36), we get
The identity implies the first asymptotics in (4.2).
Similarly, the second definition (3.4) and the second formula (2.37) give
The identity (2.21) implies
Substituting (2.33)–(2.35) into these asymptotics and using the second identity in (2.36), we get
This implies the second asymptotics in (4.2). □
Second, we determine asymptotics of the eigenvalues .
Let. Then the eigenvaluessatisfy the asymptoticswhereis defined by (
4.3
).
Let , . Lemma 10 shows
Substituting the asymptotics (4.2) into (3.3), we get
where α, β, and have the form (3.2) and (4.1), respectively. Using again (3.2) and (4.5), we get
Then
Substituting these asymptotics into (4.6) and using the asymptotics , we obtain
The equation gives
Then identity (4.5) implies
which yields the asymptotics (4.4). □
The formula (1.4) is proved in Lemma 10. Substituting the identity into asymptotics (4.4), we get (1.5). □
Trace formula
Asymptotics of the characteristic function
Now we determine asymptotics of the functions and defined by (3.4) in the case . Moreover, we assume that q is 1-periodic function, i. e. .
Let . Then it follows from the second identity in (2.36) that . Now we can integrate by parts in the integrals in the asymptotics (2.34) and (2.35). Therefore, , , , as . Substituting these asymptotics into (2.37), we get
where W is defined by (2.21). Substituting these asymptotics into (3.4), we obtain (5.1). □
In order to determine the asymptotics of the characterstic function D we use the representation (2.6).
Let. Thenasfor. Here the functions α and β given by (
3.2
) and.
Let and . Substituting (5.1) into (3.3), we obtain
where α and β have the form (3.2).
The formulas (2.21) and (2.19) imply
Substituting these asymptotics and (2.9) into (2.41) and using (2.33), we get
Substituting the last asymptotics and (5.3) into (2.6), we obtain the asymptotics (5.2). □
Trace formula
In this Section we prove the trace formula for the shifted operators given by (1.1), where , .
Let . Introduce the function , , where N is given in Lemma 5(ii), and the resolvents . Then we have
where the contours are given by
Using the identities , we get for some constant and , where , . These estimates yield .
The asymptotics (1.5) show that the series (1.6) converges absolutely and uniformly in .
Let and . The asymptotics (2.47) shows that is well defined on the contours for large by the condition . For large enough we have
Let . The asymptotics (5.2) give
for uniformly in . These formula yields
Integrating by parts we have
Equation (5.4) gives the identity (1.6). □
References
1.
E.F.Akhmerova, Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order , Math. Notes.90(5–6) (2011), 813–823. doi:10.1134/S0001434611110216.
2.
A.Badanin and E.Korotyaev, Spectral asymptotics for periodic fourth-order operator, Int. Math. Res. Not.45 (2005), 2775–2814. doi:10.1155/IMRN.2005.2775.
3.
A.Badanin and E.Korotyaev, Spectral estimates for periodic fourth order operators, St. Petersburg Math. J.22(5) (2011), 703–736. doi:10.1090/S1061-0022-2011-01164-1.
4.
A.Badanin and E.Korotyaev, Even order periodic operator on the real line, Int. Math. Res. Not.5 (2012), 1143–1194. doi:10.1093/imrn/rnr057.
5.
A.Badanin and E.Korotyaev, Trace formula for fourth order operators on the circle, DPDE.10(4) (2013), 343–352. doi:10.4310/DPDE.2013.v10.n4.a2.
6.
A.Badanin and E.Korotyaev, Sharp eigenvalue asymptotics for fourth order operators on the circle, J. Math. Anal. Appl.417 (2014), 804–818. doi:10.1016/j.jmaa.2014.03.069.
7.
A.Badanin and E.Korotyaev, Trace formula for fourth order operators on unit interval. II, DPDE12(3) (2015), 217–239. doi:10.4310/DPDE.2015.v12.n3.a2.
8.
A.Badanin and E.Korotyaev, Third-order operators with three-point conditions associated with Boussinesq’s equation, Appl. Anal.100(3) (2021), 527–560. doi:10.1080/00036811.2019.1610941.
9.
A.Bayramov, Z.Oer, S.Öztürk Uslu and S.Kizilbudak Caliskan, On the regularized trace of a fourth order regular differential equation, Int. J. Contemp. Math. Sci.1 (2006), 245–254. doi:10.12988/ijcms.2006.06025.
10.
L.F.CaudillJr., P.A.Perry and A.W.Schueller, Isospectral sets for fourth-order ordinary differential operators, SIAM J. Math. Anal.29(4) (1998), 935–966. doi:10.1137/S0036141096311198.
11.
M.V.Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations, Springer Science and Business Media, 2012.
12.
C.T.Fulton and S.A.Pruess, Eigenvalue and eigenfunction asymptotics for regular Sturm–Liouville problems, J. Math. Anal. Appl.188(1) (1994), 297–340. doi:10.1006/jmaa.1994.1429.
13.
E.Gül, The trace formula for a differential operator of fourth order with bounded operator coefficients and two terms, Turk. J. Math.28 (2004), 231–254.
14.
E.Gül and A.Ceyhan, A second regularized trace formula for a fourth order differential operator, Symmetry13(4) (2021), 629. doi:10.3390/sym13040629.
15.
H.Gunes, N.B.Kerimov and U.Kaya, Spectral properties of fourth order differential operators with periodic and antiperiodic boundary conditions, Results Math.68 (2015), 501–518. doi:10.1007/s00025-015-0454-2.
16.
B.M.Levitan and I.S.Sargsyan, Sturm–Liouville and Dirac Operators, Kluwer, Dordrecht, 1991.
17.
V.A.Marchenko, The Sturm–Liouville Operators and Their Applications, Birkhäuser Verlag, Basel, 1986.
18.
J.R.McLaughlin, An inverse eigenvalue problem of order four – an infinite case, SIAM J. Math. Anal.9(3) (1978), 395–413. doi:10.1137/0509026.
19.
V.Mikhailets and V.Molyboga, Uniform estimates for the semi-periodic eigenvalues of the singular differential operators, Meth. Funct. Anal. Topol.10(4) (2004), 30–57.
20.
M.Naimark, Linear Differential Operators. Part I. Elementary Theory of Linear Differential Operators, Frederick Ungar Publishing, New York, NY, 1967.
21.
A.I.Nazarov, D.M.Stolyarov and P.B.Zatitskiy, Tamarkin equiconvergence theorem and trace formula revisited, J. Spect. Theory4(2) (2014), 365–389. doi:10.4171/JST/73.
22.
V.G.Papanicolaou, The spectral theory of the vibrating periodic beam, Commun. Math. Phys.170 (1995), 359–373. doi:10.1007/BF02108333.
23.
V.G.Papanicolaou, Trace formulas and the behaviour of large eigenvalues, SIAM J. Math. Anal.26(1) (1995), 218–237. doi:10.1137/S0036141092224601.
D.M.Polyakov, Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients, Sib. Math. J.56(1) (2015), 138–154. doi:10.1134/S0037446615010140.
26.
D.M.Polyakov, Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions, St. Petersburg Math. J.27(5) (2016), 789–811. doi:10.1090/spmj/1417.
27.
D.M.Polyakov, Spectral properties of an even-order differential operator, Diff. Equat.52(8) (2016), 1098–1103. doi:10.1134/S0012266116080176.
28.
D.M.Polyakov, Spectral estimates for the fourth-order operator with matrix coefficients, Comp. Math. Math. Phys.60(7) (2020), 1163–1184. doi:10.1134/S0965542520050139.
29.
V.A.Sadovnichii, The trace of ordinary differential operators of high order, Math. USSR-Sb.1 (1967), 263–288. doi:10.1070/SM1967v001n02ABEH001979.
30.
V.A.Sadovnichii and V.E.Podolskii, Traces of operators, Russ. Math. Surv.61 (2006), 885–953. doi:10.1070/RM2006v061n05ABEH004357.
31.
R.F.Shevchenko, On the trace of a differential operator, Soviet Math. Dokl.6 (1965), 1183–1186.
