11institutetext: University of Massachusetts Amherst
Contact Email: 11email<EMAIL_ADDRESS>
# Algebra Error Classification
with Large Language Models

Hunter McNichols  Mengxue Zhang  Andrew Lan

###### Abstract

Automated feedback as students answer open-ended math questions has
significant potential in improving learning outcomes at large scale. A key
part of automated feedback systems is an error classification component, which
identifies student errors and enables appropriate, predefined feedback to be
deployed. Most existing approaches to error classification use a rule-based
method, which has limited capacity to generalize. Existing data-driven methods
avoid these limitations but specifically require mathematical expressions in
student responses to be parsed into syntax trees. This requirement is itself a
limitation, since student responses are not always syntactically valid and
cannot be converted into trees. In this work, we introduce a flexible method
for error classification using pre-trained large language models. We
demonstrate that our method can outperform existing methods in algebra error
classification, and is able to classify a larger set of student responses.
Additionally, we analyze common classification errors made by our method and
discuss limitations of automated error classification.

###### Keywords:

Error Classification Large Language Models Math Education

##  1 Introduction

Quality math education, particularly at a young age, is of crucial importance
for students growing into an increasingly technology-driven world. Intelligent
tutoring systems (ITSs) have demonstrated their effectiveness in improving
math learning outcomes [1, 10, 24]. Many ITSs have a component that provides
automated feedback for students while they solve math questions. This
component enables teachers to provide personalized feedback at scale, since
they can write feedback once that applies to many students. Moreover, in some
ITSs, this feedback can direct a student to the precise error in their problem
solving process. This direct, timely feedback enables immediate learning and
helps students refine their math skills at their own pace [21].

One crucial part of feedback generation is the task of error classification,
i.e., detecting student errors and the class corresponding to the error type.
Once detected, the error class then informs the feedback generator which
feedback is appropriate to provide to the student. This is especially
important for open-ended math questions, where students have to reason step-
by-step and one error can lead a student away from the correct solution. A
related line of work is automated grading of open-ended math responses [15, 9,
2, 3, 31], which is closely related to the automated short answer grading
(ASAG) and automated essay scoring (AES) tasks [20, 5, 27, 28]. However,
student error classification has an important difference from automated
scoring: the former operates at a finer-grained level, focusing on individual
solution steps, while the latter operates at a higher level, focusing on one
score for the entire response.

Traditionally, error classification systems require a domain expert to develop
hand-crafted rules that outline the different patterns of errors possible in a
question [4, 13, 11]. As a result, these systems are resource-intensive to
create and do not generalize to responses that are not foreseen by the system
designers. Data-driven methods have the potential to generalize to a wider
range of responses, but few have been explored and most methods only process a
final solution and not intermediate steps. The leading data-driven method to
intermediate error classification relies on a tree embedding method [32]. This
method is highly restrictive since it requires mathematical expressions in
student responses to be converted to syntax trees [29]. Student solutions
(particularly incorrect ones) are not always syntactically valid, which means
they cannot be processed by this method. Additionally, this method only uses
equations as input and cannot be easily extended to include other student
response information, such as natural language text or ITS-recorded
interactions.

####  1.0.1 Contributions

In this paper, we propose a method that overcomes the above limitations by
using pre-trained Large Language Models (LLMs) for algebra error
classification. Our contributions are:

  * •

We outline a method towards using pre-trained LLMs for algebra error
classification, which enables us to handle any student response, regardless of
syntactic validity.

  * •

We demonstrate the effectiveness of this method through experimental
evaluation using an algebra problem-solving dataset. We compare various pre-
trained LLMs to the existing data-driven baseline, and show that BERT (and no
other LLM) outperforms it.

  * •

We showcase the flexibility of our method in that we can easily add more
information into our classification system, which results in further
performance improvement. We show how one can augment our method by adding ITS-
recorded interactions to the input, or through Domain Adapation (DA) that
incorporates further domain-specific pre-training to the LLM.

  * •

We perform qualitative analysis on the cause of classification errors in our
method and discuss avenues of future work.

##  2 Methodology

In this section, we outline how we formulate the problem of algebra error
classification. Then, we outline our method, which uses a LLM to classify
errors.

###  2.1 The Algebra Error Classification Task

The setup of the algebra error classification problem in an open-ended math
setting is as follows. We have a set of predefined error classes for a
category of math questions $C=\\{c_{1}\dots c_{m}\\}$. We consider a solution
that a student provides to a math question as a series of steps
$S=\\{s_{1},\dots,s_{n}\\}$ (for simplicity of exposition, question statement
can be included in the first step, $s_{1}$). A step encompasses a resulting
intermediate equation and descriptions of the process by which a student comes
to this result. This description includes, but is not limited to, the
application of a theorem, transformation, or intermediate calculation. Assume
that, during step $s_{t}$, a student makes an error. We aim to identify the
type of error $c_{i}$ made by a student at $s_{t}$ given the step history
$S_{t}=\\{s_{1}...s_{t}\\}$.

Concretely, we consider a question from an introduction to algebra class:

$\displaystyle\text{Solve\ for}\ x:\ x+4=8$

With a student’s solution:

Step 1: $\displaystyle\ x+4=8$ Step 2: $\displaystyle\ x+4+4=8+4$ Step 3:
$\displaystyle\ x=12$

In this case, an error occurs at $s_{2}$, where a student adds to both sides
instead of subtracting. While this is a valid algebra, it indicates the
student is not progressing towards the solution. The task is to identify the
class $c_{i}$ which corresponds to a “wrong operation” error. As input, we are
given $S_{t}=\\{s_{1},s_{2}\\}$. A classification could also be done for the
addition error at $s_{3}$ using $S_{t}=\\{s_{1},s_{2},s_{3}\\}$.

It is important to note that step information is not limited to the equation.
It could include any additional information available about how the student
arrived at that equation. For example, notes in the margin between equations
or ITS-recorded interactions which indicate student intent.

###  2.2 Our Method

Our method to solve the algebra error classification task is to fine-tune a
pre-trained LLM, which we detail below.

####  2.2.1 Model Architecture

We experiment with two main types of LLMs: those pre-trained on the masked
language modeling (MLM) objective, such as BERT [8], and those pre-trained on
next token prediction in an autoregressive way, such as (GPT-2)[22]. Each type
requires a different setup to be used as an error classifier.

For MLM-type LLMs, the output contains the vector embedding of a [CLS] token.
This token represents the contextual information of the entire input text
sequence. As such, these models are designed for sequence classification. To
form our classification head, we connect a linear layer from the [CLS] token
to a probability space, which is the size of our number of error classes. For
autoregressive models, such as GPT-2 [22], there is no [CLS] token. To form
our classification head, we instead connect a linear layer from the final
hidden state representation (i.e. representation after the entire input
sequence is encoded) to a probability space, which is the size of our number
of error classes.

As input to the LLM, we tokenize a string that represents the step history. In
our experiments, we try different methods of representing step history and
compare the results. We note that input formulation is the most flexible part
of our architecture and one of the main advantages of this method. First, it
does not constrain the student responses to be a valid syntax tree. For
example, response of “$x+3=\text{unicorn}$” is still a valid string. Second,
it can be easily adapted to include further information. In our experiments,
we only represent information from two steps $s_{t-1}$ and $s_{t}$. This
representation is used to provide fair comparison to the leading data-driven
method, which only considers the current and prior equation trees. However,
this method could be easily extended to include the entire sequence history
$S_{t}$ or responses to prior questions.

Figure 1: Architecture diagram for our LLM-based method. Input is an arbitrary
string for a set of steps. We depict the action information described in
Section 3.1 as additional step information. Output is a class corresponding to
error detected in student response.

####  2.2.2 Task-specific training

To train our model for the task of algebra error classification, we minimize
the standard cross-entropy loss[19] for error class predictions. We
backpropagate this loss to learn the linear classification head from scratch
and fine-tune the LLM. We also experiment with performing DA prior to fine-
tuning. In DA, we perform additional pre-training on the LLM to adapt it to
the domain-specific vocabulary of our task. This is performed with a domain-
related dataset. In our case, we use a subset of the AMPS Mathematica dataset
[12] containing solutions to PEMDAS questions generated by Mathematica. This
DA step adds additional flexibility to our method, since we can readily adapt
the LLM to more complex math questions, such as calculus or probability
theory. This pre-training may make the LLM more familiar with advanced
mathematical notation and reasoning. We leave this exploration for future
work.

##  3 Experimental Evaluation

In this section, we detail our experiments and analyses. First, we show our
quantitative experiments that compare the performance of our method to the
performance of the baseline method. Then, we show quantitative experiments on
two variations of our method, which use different strategies to include
additional contextual information. Finally, we perform a qualitative error
analysis on our best performing method.

###  3.1 Dataset Details

For our experiments, we use the CogTutor dataset accessed via PSLC Datashop
[14, 23]. The dataset contains logs of students’ step-by-step responses to
questions in an Algebra I course. These logs were recorded during student
interactions with an ITS named Cognitive Tutor. In Cognitive Tutor, students
select from a set of predefined actions to manipulate an algebra equation.
After selecting an action, the student inputs the expected resulting equation.
Interactions are present in the dataset for each part of this process. We
group the action interaction and the equation writing interactions into a
single step.

The Dataset, in total, contains 130,823 interactions. These interactions span
9590 responses across 95 unique students. No demographic information is
provided about the students. From the total interactions, we use a grouped
subset of 5,744 steps that have a detected student error called BUG [32]. In
these cases, the ITS showed an automated feedback message to the student
indicating an error made during the equation-solving process. These messages
are based on a set of 92 predefined rules in the ITS. The feedback messages
vary in length from short hints, such as “Check your sign.” to more
descriptive feedback, such as “You are dividing a positive by a negative. The
result should be negative.”

####  3.1.1 Baseline Labeling.

For our initial comparisons, we use the step labeling introduced in the
leading data-driven method which has 24 distinct labels [32]. This labeling
was present for 3,318 of the 5,744 BUG steps; the remaining 2,426 steps
contained equations which were not compatible with the Tree-Embedding approach
utilized in the baseline method.

####  3.1.2 New Labeling.

In subsequent experiments, where we compare variations of our method, we
utilize our own labeling of the BUG steps, which has 19 labels on 5,339 of the
5,744 BUG steps. The remaining 403 steps were dropped from our analysis since
they were either i) system errors not related to math operations or ii) rare
errors that were present less than 30 times across the entire dataset. We make
this new dataset, called CogTutorBugs, publicly available
111https://github.com/umass-ml4ed/CogTutorBugs.

###  3.2 Metrics and Baselines

For error classification, our primary metric is classification accuracy, i.e.,
percentage of correctly predicted classes in the test set. We compare the
following:

  * •

TE+C Best performing baseline method introduced in prior work [32]. This
method converts intermediate equations in student response steps into tree-
embeddings [29], which are used to learn math operation embeddings. The
operation embeddings are then used in a supervised learning manner for error
classification.

  * •

GRU+C Additional method explored in the baseline paper. Concatenates the
equations together and passes the combined sequence character-by-character
into a GRU [6]. Final output state is used for error classification.

  * •

BERT Bidirectional encoder representations from transformers (BERT) [8]. The
predominant MLM-based LLM, with a format that is well-suited for
classification tasks.

  * •

GPT-2 Generative Pre-trained Transformer 2 (GPT-2). An autoregressive LLM pre-
trained for next sequence prediction instead of MLM. GPT-2 is well-suited for
text generation tasks, but can be used for classification [22].

  * •

MathBERT A BERT-based LLM pre-trained on a corpus of mathematics textbooks and
course material [26].

  * •

XLM-RoBERTa Multilingual variation of RoBERTa [17], a BERT-based LLM pre-
trained on specific natural language processing benchmark tasks such as
question answering, reading comprehension, and natural language understanding.
XLM-RoBERTa often outperforms BERT on these tasks [7].

###  3.3 Implementation Details

We perform five-fold cross-validation on the BUG subsets of the CogTutor
dataset. We use four folds for fine-tuning the LLMs and reserve the final fold
for calculating test accuracy. For all experiments we train our models for 30
epochs. We report the mean and standard deviation in final-epoch test accuracy
across the five folds.

All pre-trained LLMs are sourced from the HuggingFace [30] transformers
library. For fine-tuning we use the AdamW optimizer [18], batch size of 300,
and learning rate of $5\cdot 10^{-5}$. We do not perform hyperparameter tuning
and use fixed hyperparameters across all models. All models were fine-tuned
for 30 epochs. We found this number of epochs was sufficient to obtain optimal
performance. All models were fine-tuned on a single NVIDIA RTX 8000 GPU. For
each fold, fine-tuning took between 4 and 9 minutes, depending on the selected
LLM.

###  3.4 Results and Analysis

Table 1: Classification accuracy, using the error labeling in [32], for our
method with selected LLMs and the baseline methods. Bold indicates the best
result.

Method | Accuracy $\uparrow$
---|---
GRU+C | $75.35\pm 1.41$
TE+C | $78.71\pm 1.74$
BERT | $\mathbf{80.68\pm 1.1}$
GPT-2 | $71.10\pm 2.58$
MathBERT | $72.66\pm 2.21$
XLM-RoBERTa | $72.75\pm 1.54$

####  3.4.1 Comparison to Baseline Methods.

We report the accuracy of our method with each LLM and the baseline methods in
Table 1. We observe that BERT has the best performance across all LLMs and is
a slight improvement over the leading baseline method (TE+C). Moreover, it
achieves this performance without needing the tree-embeddings of the response
equations used in TE+C. Perhaps surprisingly, other LLMs do not perform as
well as either baseline method. This observation suggests that the choice of
the LLM architecture and initial weights configuration is crucial for the
success of LLMs in classification tasks. BERT appears to be more robust in
fine-tuning for the specific task of algebra error classification. On the
contrary, GPT-2 isn’t well-suited for classification likely due to its
autoregressive design. XLM-RoBERTa and MathBERT are pre-trained on other tasks
and show limited ability to generalize to our task. Their less competitive
performance suggests that pre-training an LLM for one task may degrade
performance in another, completely different task. We find it surprising that
MathBERT, which is specifically designed for mathematical content, does not
perform well on our task. One possible explanation is that MathBERT was pre-
trained on textbook content, which can be different from problem-solving
content. It also is pre-trained on student work in other ITSs, which may
differ in structure to Cognitive Tutor.

Table 2: Classification accuracy, using our new error labeling, for our method
with selected LLMs. Control is the accuracy only with equation information in
the input. Action Inc. is the accuracy with additional student action context
provided by Cognitive Tutor ITS logs. Bold indicates the best result.

Language Model | Control $\uparrow$ | Action Inc. $\uparrow$
---|---|---
BERT | $82.02\pm 1.09$ | $\mathbf{85.90\pm 1.86}$
GPT-2 | $76.46\pm 1.47$ | $74.21\pm 2.08$
MathBERT | $76.08\pm 2.05$ | $81.27\pm 0.32$
XLM-RoBERTa | $77.73\pm 2.59$ | $84.23\pm 2.65$

####  3.4.2 Use of Additional Context.

An advantage in using a LLM-based method for the error classification task is
the flexibility of the input. Therefore, we experiment with including
information about the student-selected action in the input (in addition to the
step equations). This action information is recorded by Cognitive Tutor and
provides context on student intent. We report the classification accuracy for
the selected LLMs in Table 2. For these experiments, we use our new feedback
labeling. We observe that introducing action information improves the
performance of all MLM models significantly. XLM-RoBERTa, in particular, has
the largest improvement of 6.5%. We note that GPT-2 actually shows a small
performance deterioration when action information is included. This
deterioration is small and could be explained by random noise in the
experiments. Another potential explanation is that GPT-2 is pre-trained on
natural language text, but the action information is programmatic, system log
text.

Table 3: Classification accuracy for BERT with various degrees of Domain
Adaptation (DA). For example, BERT + 3 Epoch DA indicates 3 epochs of DA
before classification fine-tuning and testing.

Model | Accuracy $\uparrow$
---|---
BERT | $82.02\pm 1.09$
BERT + 3 epoch DA | $81.10\pm 2.40$
BERT + 10 epoch DA | $81.00\pm 1.72$

Figure 2: Test accuracy comparison of DA methods. We observe only a slight
improvement in initial epochs with DA and no significant difference in final
performance (compared to no DA). The graph only shows first 15 epochs for
clearer observation of early epochs. Shaded regions indicate range of accuracy
across all folds. Best viewed in color.

####  3.4.3 Domain Adaptation

LLMs, such as BERT and GPT-2, are pre-trained on a variety of textual data but
do not have domain-specific vocabulary for middle-school-level math. To adapt
the LLM to better understand domain-specific vocabulary, we perform additional
pre-training with in-domain but not task-related data. This step is done prior
to task-specific fine-tuning and often called Domain Adaptation (DA). In Table
3, we detail our experiments on performing DA with the PEMDAS subset of the
AMPS Mathmatica dataset [12]. We observe no significant change in overall
accuracy with the introduction of either 3 or 10 epochs of DA. One possible
explanation is the small vocabulary size of our domain: only numbers and
fundamental math operations: $+,-,/,*,=,()$. However, we do see a non-trivial
performance improvement in initial epochs of training as depicted in Figure 2.
This observation suggests that DA does provide some context on basic math
vocabulary and mathematical reasoning, but this context is sufficiently
learned from task fine-tuning within the first few epochs.

Figure 3: Mis-classification heatmap for qualitative error analysis of our
method. Value in each cell is the ratio of the predicted class count to total
number of actual class instances in the test set (each row sums to 1). Best
viewed in color. Table 4: Example feedback templates for classes described in
qualitative error analysis. The actual feedback students see have question-
specific values for $A$, $B$, and $expression$.

# | Class Label |  Sample Feedback
---|---|---
4 | REVERSED_SIDES |  “$expression$ is equal to $A$ minus $B$. You need to calculate $-A$ minus $B$.”
8 | WRONG_OPERATION |  “$expression$ is equal to $A$ plus $B$. You need to calculate $A$ minus $B$.”
12 | FORGOT_NEGATIVE |  “You forgot the negative sign.”
15 | CHECK_YOUR_SIGN |  “$expression$ Check your sign.”
3 | MULTIPLY_TO_SIMPLIFY |  “Put the expression in its simplest form by performing multiplication on the right side.”
13 | SIMPLIFY_FRACTION |  “Simplify the fraction $expression$”

####  3.4.4 Qualitative Error Analysis

We now analyze the cases when our best trained method (BERT + Action) mis-
classifies student errors in a sample cross-validation fold. We show a mis-
classification heatmap in Figure 3. We observe two major causes of mis-
classification: between two error classes where intent is ambiguous and
between classes that are equivalent, but vary in level of specificity.
Detailed information about each class mentioned is provided in Table 4.

The most frequent mis-classification is between error class #4 and #8
(REVERSED_SIDES and WRONG_OPERATION respectively). Class #4, 41% of the time,
is incorrectly classified as class #8. 18% of the time the reverse mis-
classification occurs. We observe that, in some scenarios, a REVERSED_SIDES
error is equivalent to a WRONG_OPERATION error. Consider the question $y+5=-2$
and a student response $y=-3$. The student could have attempted to subtract 5
from both sides, but dropped the negative on -2. Equivalently, the student
could have attempted to subtract -5 from both sides, erroneously thinking
subtraction of -5 would cancel the 5 on the left hand side and confusing the
rules of subtracting negatives. Both error classes can be appropriate,
depending on the student’s thought process. However, the true student intent
is unknown, so it’s ambiguous which class is correct.

Other frequent mis-classifications occur between classes that are almost the
same but different in level of specificity. For example, class #12
(FORGOT_NEGATIVE) is 23.6% of the time mis-classified as class #8
(WRONG_OPERATION). In the case of addition and subtraction, a sign error can
be explained by the selection of an incorrect operation. In that case, an
operation error is a more specific error than the sign error. A similar type
of mis-classification occurs 13.2% of the time from class #15 to class #18,
and 11% of the time from class #3 to class #13.

Through this qualitative analysis, we observe the most frequent mis-
classifications of our method are explainable. They often occur because of
ambiguous or overlapping classes. These observations suggest our method may be
performing better than the accuracy metrics indicate. The observations also
suggest that there is an upper limit to the performance possible on our chosen
dataset, depending on how error classes are defined. Furthermore, the feedback
selected by the rules of the Cognitive Tutor ITS should not be seen as the
absolute truth. Without a perfect knowledge of student intent for each step of
a response, it is impossible to have certainty about the true cause of student
error. This is a general limitation which underlies all error classification
systems.

##  4 Discussion, Conclusion, and Future Work

In this paper, we detailed a large language model-based method for the task of
math error classification in open-ended questions. We now discuss our key
observations and outline avenues for future work. First, we observe that our
method, when combined with BERT, performs better than the best-performing,
data-driven method for math error classification. Additionally, since it does
not rely on syntactically correct responses, the method can operate on a wider
range of student responses than the baseline method. Second, we observe that
that incorporating additional information about student intent is easily
achievable with our method and can provide a significant performance
improvement. We hypothesize that providing further contextual information,
such as student knowledge levels, will continue to improve performance, e.g.,
using models that can understand open-ended responses [16, 25]. Third, we
observe, through our qualitative analysis, that the errors made by our method
are mainly due to ambiguous error class labels. This observation suggests an
avenue for future exploration, where we extend our method to generate
feedback, rather than classify errors.

##  5 Acknowledgements

The authors thank the NSF (grants 1917713, 2118706, 2202506, 2215193) for
partially supporting this work.

## References

  * [1] Aleven, V., McLaughlin, E.A., Glenn, R.A., Koedinger, K.R.: Instruction based on adaptive learning technologies. Handbook of research on learning and instruction pp. 522–560 (2016)
  * [2] Baral, S., Botelho, A.F., Erickson, J.A., Benachamardi, P., Heffernan, N.T.: Improving automated scoring of student open responses in mathematics. International Educational Data Mining Society (2021)
  * [3] Baral, S., Seetharaman, K., Botelho, A.F., Wang, A., Heineman, G., Heffernan, N.T.: Enhancing auto-scoring of student open responses in the presence of mathematical terms and expressions. In: International Conference on Artificial Intelligence in Education. pp. 685–690. Springer (2022)
  * [4] Brusilovsky, P., Peylo, C.: Adaptive and intelligent web-based educational systems. International Journal of Artificial Intelligence in Education 13(2-4), 159–172 (2003)
  * [5] Burstein, J.: The e-rater® scoring engine: Automated essay scoring with natural language processing. (2003)
  * [6] Chung, J., Gulcehre, C., Cho, K., Bengio, Y.: Empirical evaluation of gated recurrent neural networks on sequence modeling. arXiv preprint arXiv:1412.3555 (2014)
  * [7] Conneau, A., Khandelwal, K., Goyal, N., Chaudhary, V., Wenzek, G., Guzmán, F., Grave, E., Ott, M., Zettlemoyer, L., Stoyanov, V.: Unsupervised cross-lingual representation learning at scale. arXiv preprint arXiv:1911.02116 (2019)
  * [8] Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: Bert: Pre-training of deep bidirectional transformers for language understanding (2018), https://arxiv.org/abs/1810.04805
  * [9] Erickson, J.A., Botelho, A.F., McAteer, S., Varatharaj, A., Heffernan, N.T.: The automated grading of student open responses in mathematics. In: International Conference on Learning Analytics & Knowledge. p. 615–624 (2020)
  * [10] Fancsali, S.E., Ritter, S.: Context personalization, preferences, and performance in an intelligent tutoring system for middle school mathematics. In: International conference on learning analytics and knowledge. pp. 73–77 (2014)
  * [11] Heeren, B., Jeuring, J., Sosnovsky, S.A., Drijvers, P., Boon, P., Tacoma, S., Koops, J., Weinberger, A., Grugeon-Allys, B., Chenevotot-Quentin, F., van Wijk, J., van Walree, F.: Fine-grained cognitive assessment based on free-form input for math story problems. In: Pammer-Schindler, V., Pérez-Sanagustín, M., Drachsler, H., Elferink, R., Scheffel, M. (eds.) Lifelong Technology-Enhanced Learning - 13th European Conference on Technology Enhanced Learning, EC-TEL 2018, Leeds, UK, September 3-5, 2018, Proceedings. Lecture Notes in Computer Science, vol. 11082, pp. 262–276. Springer (2018). https://doi.org/10.1007/978-3-319-98572-5_20, https://doi.org/10.1007/978-3-319-98572-5_20
  * [12] Hendrycks, D., Burns, C., Kadavath, S., Arora, A., Basart, S., Tang, E., Song, D., Steinhardt, J.: Measuring mathematical problem solving with the math dataset. NeurIPS (2021)
  * [13] Koedinger, K.R., Anderson, J.R., Hadley, W.H., Mark, M.A.: Intelligent tutoring goes to school in the big city. International Journal of Artificial Intelligence in Education 8, 30–43 (1997)
  * [14] Koedinger, K.R., Baker, R.S., Cunningham, K., Skogsholm, A., Leber, B., Stamper, J.: A data repository for the edm community: The pslc datashop. Handbook of educational data mining 43, 43–56 (2010)
  * [15] Lan, A.S., Vats, D., Waters, A.E., Baraniuk, R.G.: Mathematical language processing: Automatic grading and feedback for open response mathematical questions. In: Proceedings of the ACM conference on learning@scale. pp. 167–176 (2015)
  * [16] Liu, N., Wang, Z., Baraniuk, R., Lan, A.: Open-ended knowledge tracing for computer science education. In: Conference on Empirical Methods in Natural Language Processing. pp. 3849–3862 (2022)
  * [17] Liu, Y., Ott, M., Goyal, N., Du, J., Joshi, M., Chen, D., Levy, O., Lewis, M., Zettlemoyer, L., Stoyanov, V.: Roberta: A robustly optimized bert pretraining approach (2019). https://doi.org/10.48550/ARXIV.1907.11692, https://arxiv.org/abs/1907.11692
  * [18] Loshchilov, I., Hutter, F.: Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101 (2017)
  * [19] Murphy, K.P.: Machine learning: A probabilistic perspective. MIT Press (2021)
  * [20] Page, E.B.: The imminence of grading essays by computer. The Phi Delta Kappan 47(5), 238–243 (1966)
  * [21] Pane, J.F., Griffin, B.A., McCaffrey, D.F., Karam, R.: Effectiveness of cognitive tutor algebra i at scale. Educational Evaluation and Policy Analysis 36(2), 127–144 (2014)
  * [22] Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., Sutskever, I., et al.: Language models are unsupervised multitask learners. OpenAI blog 1(8), 9 (2019)
  * [23] Ritter, S., Anderson, J.R., Koedinger, K.R., Corbett, A.: Cognitive tutor: Applied research in mathematics education. Psychonomic bulletin & review 14(2), 249–255 (2007)
  * [24] Roschelle, J., Feng, M., Murphy, R.F., Mason, C.A.: Online mathematics homework increases student achievement. AERA open 2(4), 2332858416673968 (2016)
  * [25] Scarlatos, A., Lan, A.: Tree-based representation and generation of natural and mathematical language. In: Association for Computational Linguistics (ACL) (2023, preprint: https://arxivorg/abs/230207974)
  * [26] Shen, J.T., Yamashita, M., Prihar, E., Heffernan, N.T., Wu, X., Lee, D.: Mathbert: A pre-trained language model for general NLP tasks in mathematics education. CoRR abs/2106.07340 (2021), https://arxiv.org/abs/2106.07340
  * [27] Uto, M., Uchida, Y.: Automated short-answer grading using deep neural networks and item response theory. In: International Conference on Artificial Intelligence in Education. pp. 334–339 (2020)
  * [28] Wang, Z., Lan, A., Waters, A., Grimaldi, P., Baraniuk, R.: A meta-learning augmented bidirectional transformer model for automatic short answer grading. In: Proc. 12th Int. Conf. Educ. Data Mining (EDM). pp. 1–4 (2019)
  * [29] Wang, Z., Lan, A.S., Baraniuk, R.G.: Mathematical formula representation via tree embeddings. In: iTextbooks@ AIED. pp. 121–133 (2021)
  * [30] Wolf, T., Debut, L., Sanh, V., Chaumond, J., Delangue, C., Moi, A., Cistac, P., Rault, T., Louf, R., Funtowicz, M., Davison, J., Shleifer, S., von Platen, P., Ma, C., Jernite, Y., Plu, J., Xu, C., Scao, T.L., Gugger, S., Drame, M., Lhoest, Q., Rush, A.M.: Huggingface’s transformers: State-of-the-art natural language processing (2019). https://doi.org/10.48550/ARXIV.1910.03771, https://arxiv.org/abs/1910.03771
  * [31] Zhang, M., Baral, S., Heffernan, N., Lan, A.: Automatic short math answer grading via in-context meta-learning. arXiv preprint arXiv:2205.15219 (2022)
  * [32] Zhang, M., Wang, Z., Baraniuk, R.G., Lan, A.S.: Math operation embeddings for open-ended solution analysis and feedback. CoRR abs/2104.12047 (2021), https://arxiv.org/abs/2104.12047


111 Mathematics Subject Classification (2020). Primary: 34L40; Secondary:
34A55. Keywords: Dirac operator, spectrum, periodic boundary conditions,
inverse problem,

Characterization of the periodic and antiperiodic spectra

of non-self-adjoint Dirac operators

Alexander Makin

> ABSTRACT. The necessary and sufficient conditions are given for a sequence
> of complex numbers to be the periodic (or antiperiodic) spectrum of non-
> self-adjoint Dirac operator.

1\. Intoduction

One of the important classes of inverse spectral problems is the problem of
recovering a system of differential equations from spectral data. The solution
of such problems are considered in many papers [12,18, 29-35, and the
references therein]. The most studied are such problems for Dirac and Dirac
type differential operator. In particular, such problems for canonical Dirac
system on a finite interval

$B\mathbf{y}^{\prime}+V\mathbf{y}=\lambda\mathbf{y},$ $None$

where $\mathbf{y}={\rm col}(y_{1}(x),y_{2}(x))$,

$B=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad
V(x)=\begin{pmatrix}p(x)&q(x)\\\ q(x)&-p(x)\end{pmatrix},$

in selfadjoint case have been studied in detail. In the cases of the Dirichlet
and the Newmann boundary conditions reconstruction of a continuous potential
from two spectra was carried out in [6], from one spectrum and the norming
constants in [5], and from the spectral function in [15]. The analogous
results for Dirac operator with summable potentials were established in [1].
The case of more general separated boundary conditions was considered in [3].
In the case of unseparated boundary conditions (including periodic,
antiperiodic and quasiperiodic conditions) the indicated problem was solved in
[17, 19-22]. In non-self-adjoint case the problem of reconstructing the
potential $V(x)$ from spectral data is much more complicated, since many
methods successfully used to study selfadjoint operators are inapplicable. For
example, the characterization of the spectra of the periodic (antiperiodic)
problem for operator (1.1) with real coefficients is given in [17] in terms of
special conformal mappings, which do not exist for complex-valued potentials.
The property that the eigenvalues of corresponding Dirichlet problem and
Neumann problem are interlaced, which is often used to prove the solvability
of the basic equation, loses its meaning in the complex case. Non-self-adjoint
inverse problems for system (1.1) with different types of boundary conditions
with sufficiently smooth coefficients, which, however, could have
singularities were investigated in [2,8,24,28].

Questions of uniqueness in inverse problems for operators of type (1.1) on a
finite interval were studied in a lot of papers. In particular, uniqueness of
the inverse problem for general Dirac-type systems of order 2n was established
in [13, 14]. Also inverse theory was intensively developed for Dirac-type
operators on the axis and semiaxis by many authors. New inverse approach to
such differential systems on the semiaxis based on A-function concept was
recently considered in [7].

The aim of this paper is to find necessary and sufficient conditions of
solvability of the periodic (antiperiodic) inverse spectral problem for system
(1.1) with a nonsmooth complex-valued potential $V(x)$.

The paper is organized as follows. Section 2 contains some basic facts and
definitions related to the considered problems. In section 3 by using a
modified version of the Gelfand-Levitan-Marchenko method we prove solvability
of the basic equation and establish necessary and sufficient conditions for an
entire function to be the characteristic determinant of the considered
problem. Further, we obtain necessary and sufficient conditions for a set of
complex numbers to be the spectrum of the mentioned problem.

2\. Preliminaries

In the present paper, we consider system (1.1), where complex-valued functions
$p,q\in L_{2}(0,\pi)$ $(V\in L_{2})$ with periodic (antiperiodic) boundary
conditions

$y_{1}(0)-(-1)^{\theta}y_{1}(\pi)=0,\quad y_{2}(0)-(-1)^{\theta}y_{2}(\pi)=0,$
$None$

where $\theta=0,1$. In what follows, we introduce the Euclidean norm
$\|f\|=(|f_{1}|^{2}+|f_{2}|^{2})^{1/2}$ for vectors $f={\rm
col}(f_{1},f_{2})\in\mathbb{C}^{2}$, and set $\langle
f,g\rangle=f_{1}g_{1}+f_{2}g_{2}$. If $W$ is $2\times 2$-matrix, then we set
$\|W\|=\sup_{\|f\|=1}\|Wf\|$ and denote by $L_{2,2}(a,b)$ and
$L_{2,2}^{2,2}(a,b)$, respectively, the spaces of 2-coordinate vector
functions $f(t)={\rm col}(f_{1}(t),f_{2}(t))$ and $2\times 2$ matrix functions
$W(t)$ with finite norms

$\|f\|_{L_{2,2}(a,b)}=\left(\int_{a}^{b}\|f(t)\|^{2}dt\right)^{1/2},\quad\|W\|_{L_{2,2}^{2,2}(a,b)}=\left(\int_{a}^{b}\|W(t)\|^{2}dt\right)^{1/2}.$

The operator $\mathbb{L}\mathbf{y}=B\mathbf{y}^{\prime}+V\mathbf{y}$ is
regarded as a linear operator in the space $L_{2,2}(0,\pi)$ with the domain
$D(\mathbb{L})=\\{\mathbf{y}\in W_{1}^{1}[0,\pi]\times
W_{1}^{1}[0,\pi]:\,\mathbb{L}\mathbf{y}\in L_{2,2}(0,\pi)$,
$U_{j}(\mathbf{y})=0$ $(j=1,2)\\}$.

Denote by

$E(x,\lambda)=\begin{pmatrix}c_{1}(x,\lambda)&-s_{2}(x,\lambda)\\\
s_{1}(x,\lambda)&c_{2}(x,\lambda)\end{pmatrix}$ $None$

the matrix of the fundamental solution system to equation (1.1) with boundary
condition $E(0,\lambda)=I$, where $I$ is the unit matrix, and by
$E_{0}(x,\lambda)$ the fundamental solution system to the equation
$B\mathbf{y}^{\prime}=\lambda\mathbf{y}$ with boundary condition
$E_{0}(0,\lambda)=I$. Obviously,

$E_{0}(x,\lambda)=\begin{pmatrix}\cos\lambda x&-\sin\lambda x\\\ \sin\lambda
x&\cos\lambda x\end{pmatrix}.$

Denote the second column of the matrix $E_{0}(x,\lambda)$

$Y_{0}(x,\lambda)=\begin{pmatrix}-\sin\lambda x\\\ \cos\lambda
x\end{pmatrix}.$

It is well known that the entries of the matrix $E(x,\lambda)$ are related by
the identity

$c_{1}(x,\lambda)c_{2}(x,\lambda)+s_{1}(x,\lambda)s_{2}(x,\lambda)=1,$ $None$

which is valid for any $x,\lambda$. The matrix $E(\pi,\lambda)$ is called the
monodromy matrix of operator $\mathbb{L}\mathbf{y}$. For its entries we
introduce the notation $c_{j}(\lambda)=c_{j}(\pi,\lambda)$,
$s_{j}(\lambda)=s_{j}(\pi,\lambda)$, $j=1,2$. We denote also the class of
entire functions $f(z)$ of exponential type $\leq\sigma$ such that
$\|f\|_{L_{2}(R)}<\infty$ by $PW_{\sigma}$ . It is known [26] that the
functions $c_{j}(\lambda),s_{j}(\lambda)$ admit the representation

$c_{j}(\lambda)=\cos\pi\lambda+g_{j}(\lambda),\quad
s_{j}(\lambda)=\sin\pi\lambda+h_{j}(\lambda),$ $None$

where $g_{j},h_{j}\in PW_{\pi}$, $j=1,2$. For functions of type (2.4) the
following statement is true:

Lemma 1 [17] . Functions $u(\lambda)$ $v(\lambda)$ admit the representations

$u(\lambda)=\sin\pi\lambda+h(\lambda),\quad
v(\lambda)=\cos\pi\lambda+g(\lambda),$

where $h,g\in PW_{\pi}$, if and only if

$u(\lambda)=-\pi(\lambda_{0}-\lambda)\prod_{n=-\infty\atop n\neq
0}^{\infty}\frac{\lambda_{n}-\lambda}{n},$

where $\lambda_{n}=n+\epsilon_{n},\\{\epsilon_{n}\\}\in l_{2}$,

$v(\lambda)=\prod_{n=-\infty}^{\infty}\frac{\lambda_{n}-\lambda}{n-1/2},$

where $\lambda_{n}=n-1/2+\kappa_{n},\\{\kappa_{n}\\}\in l_{2}$.

It is well known that the characteristic determinant of problem (1.1), (2.1)
can be reduced to the form

$\Delta(\lambda)=(-1)^{\theta+1}+\frac{c_{1}(\lambda)+c_{2}(\lambda)}{2},$
$None$

nd the eigenvalues are specified by the asymptotic formulas

$\lambda_{n,j}=2n+\theta+\varepsilon_{n,j},$ $None$

where $\\{\varepsilon_{n,j}\\}\in l_{2}$, $n\in\mathbb{Z}$, $j=1,2$. Further
$\Gamma(z,r)$ denotes a disk of radius $r$ centered at the point $z$.

Next, we establish the necessary and sufficient conditions that an entire
function must satisfy in order to be the characteristic determinant of some
problem (1.1), (2.1). Then, we give an intrinsic description of sequences
which are periodic or antiperiodic spectra of operator (1.1), (2.1).

3\. Main Results

3.1. Characteristic determinant

Theorem 3.1. For a function $U(\lambda)$ to be the characteristic determinant
of problem (1.1), (2.1), it is necessary and sufficient that it can be
represented in the form

$U(\lambda)=(-1)^{\theta+1}+\cos\pi\lambda+f(\lambda),$

where $f\in PW_{\pi}$, and

$\sum_{n=-\infty}^{\infty}|f(n)|<\infty.$ $None$

Proof. Necessity. Evidently, relations (2.4), (2.5) imply that $f\in
PW_{\pi}$. To check inequality (3.1) we consider the monodromy matrix of
problem (1.1), (2.1). Let the corresponding function $s_{2}(\lambda)$ have the
roots $\lambda_{n}$, hence by [27, Lemma 2.2],

$\lambda_{n}=n+\delta_{n},$ $None$

where $\\{\delta_{n}\\}\in l_{2}$, $n\in\mathbb{Z}$ . Since

$c_{j}(\lambda_{n})=\cos\pi\lambda_{n}+g_{j}(\lambda_{n}),$ $None$

it follows from (2.4) and [27, Lemma 2.1] that

$\sum_{n=-\infty}^{\infty}|g_{j}(\lambda_{n})|^{2}<\infty.$ $None$

Denote

$\chi(\lambda)=U(\lambda)-(-1)^{\theta+1}=\cos\pi\lambda+f(\lambda).$ $None$

By virtue of (2.5),

$c_{1}(\lambda_{n})+c_{2}(\lambda_{n})=2\chi(\lambda_{n}).$

It follows from (2.3) that $c_{1}(\lambda_{n})c_{2}(\lambda_{n})=1,$
consequently the numbers $c_{1}(\lambda_{n}),c_{2}(\lambda_{n})$ are the roots
of the quadratic equation

$w^{2}-2\chi(\lambda_{n})w+1=0.$ $None$

Therefore we have

$\begin{array}[]{c}c_{1}(\lambda_{n}),c_{2}(\lambda_{n})=\chi(\lambda_{n})\pm\sqrt{\chi^{2}(\lambda_{n})-1}\\\
=\cos\pi\lambda_{n}+f(\lambda_{n})\pm\sqrt{(\cos\pi\lambda_{n}+f(\lambda_{n}))^{2}-1}\\\
=\cos\pi\lambda_{n}+f(\lambda_{n})\pm\sqrt{\cos^{2}\pi\lambda_{n}+2\cos\pi\lambda_{n}f(\lambda_{n})+f^{2}(\lambda_{n})-1}\\\
=\cos\pi\lambda_{n}+f(\lambda_{n})\pm\sqrt{2\cos\pi\lambda_{n}f(\lambda_{n})+f^{2}(\lambda_{n})-\sin^{2}\pi\lambda_{n}}.\end{array}$
$None$

It follows from (3.3) and (3.7) that

$(g_{1}(\lambda_{n})-f(\lambda_{n}))^{2}=2\cos\pi\lambda_{n}f(\lambda_{n})+f^{2}(\lambda_{n})-\sin^{2}\pi\lambda_{n},$

hence,

$2\cos\pi\lambda_{n}f(\lambda_{n})=g_{1}^{2}(\lambda_{n})-2g_{1}(\lambda_{n})f(\lambda_{n})+\sin^{2}\pi\delta_{n}.$
$None$

It follows from (3.2) that for all sufficiently large $|n|$ the inequality
$|\cos\pi\lambda_{n}|>1/2$ holds. This, together with (3.2), (3.4), and [27,
Lemma 2.1] implies

$\sum_{n=-\infty}^{\infty}|f(\lambda_{n})|<\infty.$ $None$

Since $f^{\prime}\in PW_{\pi}$, then

$|f(n)|\leq|f(\lambda_{n})|+|f(n)-f(\lambda_{n})|\leq|f(\lambda_{n})|+|\delta_{n}||\tau_{n}|\leq|f(\lambda_{n})|+(|\delta_{n}|^{2}+|\tau_{n}|^{2})/2,$

where

$\tau_{n}=\max_{\lambda\in\Gamma(n,|\delta_{n}|)}|f^{\prime}(\lambda)|.$

By [27, Lemma 2.1], $\\{\tau_{n}\\}\in l_{2}$. This and (3.9) imply (3.1).

Sufficiency. Let $f\in PW_{\pi}$ satisfy condition (3.1). It follows from the
Paley-Wiener theorem and [16, Lemma 1.3.1] that

$\lim_{|\lambda|\to\infty}e^{-|\pi{\rm Im}\lambda|}f(\lambda)=0,$ $None$

hence there exists a positive integer $N_{0}$ large enough that
$|f(\lambda)|<1/100$ if ${\rm Im}\lambda=0$, $|{\rm Re}\lambda|\geq N_{0}$.
Let $\lambda_{n}$ $(n\in\mathbb{Z})$ be a strictly monotone increasing
sequence of real numbers such that for any $n\neq 0$
$\lambda_{n}=\lambda_{-n}$, $|\lambda_{n}-(N_{0}+1/2)|<1/100$ if $0\leq n\leq
N_{0}$, and $\lambda_{n}=n$ if $n>N_{0}$. Denote

$s(\lambda)=-\pi(\lambda_{0}-\lambda)\prod_{n=-\infty\atop n\neq
0}^{\infty}\frac{\lambda_{n}-\lambda}{n}.$ $None$

It follows from Lemma 1 that

$s(\lambda)=\sin\pi\lambda+h(\lambda),$ $None$

where $h\in PW_{\pi}$, hence,

$|s(\lambda)|\geq C_{1}e^{\pi|{\rm Im}\lambda|}$ $None$

if $|{\rm Im}\lambda|\geq M$, where $M$ is sufficiently large. It follows from
(3.11) that

$\dot{s}(\lambda_{0})=\pi\prod_{n=-\infty\atop n\neq
0}^{\infty}\frac{\lambda_{n}}{n}>0.$

One can readily see that the inequality
$\dot{s}(\lambda_{n})\dot{s}(\lambda_{n+1})<0$ holds for all $n\in\mathbb{Z}$.
It follows from two last inequalities that

$(-1)^{n}\dot{s}(\lambda_{n})>0.$ $None$

Relation (3.12) and [27, Lemma 2.1] imply that

$\dot{s}(\lambda_{n})=\pi(-1)^{n}+\tau_{n},$ $None$

where $\\{\tau_{n}\\}\in l_{2}$, hence,

$\frac{1}{\dot{s}(\lambda_{n})}=\frac{(-1)^{n}}{\pi}+\sigma_{n},$ $None$

where $\\{\sigma_{n}\\}\in l_{2}$.

Equation (3.6) has the roots

$\begin{array}[]{c}c_{n}^{\pm}=\chi(\lambda_{n})\pm\sqrt{\chi^{2}(\lambda_{n})-1}=\cos\pi\lambda_{n}+f(\lambda_{n})\pm\sqrt{(\cos\pi\lambda_{n}+f(\lambda_{n}))^{2}-1}=\\\
=\cos\pi\lambda_{n}+f(\lambda_{n})\pm\sqrt{\cos^{2}\pi\lambda_{n}+2\cos\pi\lambda_{n}f(\lambda_{n})+f^{2}(\lambda_{n})-1}=\\\
=\cos\pi\lambda_{n}+f(\lambda_{n})\pm\sqrt{2\cos\pi\lambda_{n}f(\lambda_{n})+f^{2}(\lambda_{n})-\sin^{2}\pi\lambda_{n}}.\end{array}$
$None$

It follows from (3.17) that if $0<|n|\leq N$ the numbers $c_{n}^{+}$ are
contained within the disk $\Gamma(i,1/10)$, the numbers $c_{n}^{-}$ are
contained within the disk $\Gamma(-i,1/10)$, and if $|n|>N$ the numbers
$c_{n}^{\pm}$ are contained within the disk $\Gamma(1,1/10)$ for even $n$, the
numbers $c_{n}^{\pm}$ are contained within the disk $\Gamma(-1,1/10)$ for odd
$n$. Denote $c_{n}=c_{n}^{+}$ for even $n$ and $c_{n}=c_{n}^{-}$ for odd $n$.
Denote also

$z_{n}=\frac{c_{n}}{\dot{s}(\lambda_{n})}.$

It follows from (2.20) that the numbers $z_{n}$ lie strictly above the line
$l:{\rm Im}\lambda=-{\rm Re}\lambda$.

Evidently,

$\lambda_{n}=n+\rho_{n},$ $None$

where $\\{\rho_{n}\\}\in l_{2}$. It follows from (3.17) and (3.18) that

$c_{n}=(-1)^{n}+\vartheta_{n},$ $None$

where $\\{\vartheta_{n}\\}\in l_{2}$. Let
$\beta_{n}=c_{n}-\cos\pi\lambda_{n}$, then $\\{\beta_{n}\\}\in l_{2}$. Let us
consider the function

$g(\lambda)=s(\lambda)\sum_{n=-\infty}^{\infty}\frac{\beta_{n}}{\dot{s}(\lambda_{n})(\lambda-\lambda_{n})}.$

By [9, p. 120] the function $g\in PW_{\pi}$ and $g(\lambda_{n})=\beta_{n}$.
Denote $c(\lambda)=\cos\pi\lambda+g(\lambda)$, then $c(\lambda_{n})=c_{n}\neq
0$, hence, the functions $s(\lambda)$ and $c(\lambda)$ have disjoint zero
sets.

Denote

$F(x,t)=\sum_{n=-\infty}^{\infty}\left(\frac{c_{n}}{\dot{s}(\lambda_{n})}(Y_{0}(x,\lambda_{n})Y_{0}^{T}(t,\lambda_{n}))-\frac{1}{\pi}Y_{0}(x,n)Y_{0}^{T}(t,n)\right).$

It follows from [26] that

$\|F(\cdot,x)\|_{L_{2,2}^{2,2}(0,\pi)}+\|F(x,\cdot)\|_{L_{2,2}^{2,2}(0,\pi)}<C_{2},$

where $C_{2}$ not depending on $x$.

Proposition 1. For every $x\in[0,\pi]$ the homogeneous equation

$f^{T}(t)+\int_{0}^{x}f^{T}(s)F(s,t)ds=0,$ $None$

where $f(t)={\rm col}(f_{1}(t),f_{2}(t))$, $f\in L_{2,2}(0,x)$, $f(t)=0$ if
$x<t\leq\pi$ has the trivial solution only.

Multiplying equation (3.19) by $\overline{f^{T}(t)}$ and integrating the
resulting equation over segment $[0,x]$, we obtain

$\|f\|_{L_{2,2}(0,x)}^{2}+\int_{0}^{x}\langle\int_{0}^{x}f^{T}(s)F(s,t)ds,f^{T}(t)\rangle
dt=0.$

Simple computations show

$\begin{array}[]{c}f^{T}(s)F(s,t)=\sum_{n=-\infty}^{\infty}\\{z_{n}[f_{1}(s)\sin\lambda_{n}s\sin\lambda_{n}t-f_{2}(s)\cos\lambda_{n}s\sin\lambda_{n}t,\\\
-f_{1}(s)\sin\lambda_{n}s\cos\lambda_{n}t+f_{2}(s)\cos\lambda_{n}s\cos\lambda_{n}t]\\\
-\frac{1}{\pi}[f_{1}(s)\sin ns\sin nt-f_{2}(s)\cos ns\sin nt,\\\ -f_{1}(s)\sin
ns\cos nt+f_{2}(s)\cos ns\cos nt]\\}\\\
=\sum_{n=-\infty}^{\infty}\\{z_{n}[f_{1}(s)\sin\lambda_{n}s\sin\lambda_{n}t-f_{2}(s)\cos\lambda_{n}s\sin\lambda_{n}t]\\\
-\frac{1}{\pi}[f_{1}(s)\sin ns\sin nt-f_{2}(s)\cos ns\sin nt],\\\
z_{n}[-f_{1}(s)\sin\lambda_{n}s\cos\lambda_{n}t+f_{2}(s)\cos\lambda_{n}s\cos\lambda_{n}t]\\\
-\frac{1}{\pi}[-f_{1}(s)\sin ns\cos nt+f_{2}(s)\cos ns\cos nt]\\},\end{array}$

therefore, transforming the iterated integrals into products of integrals and
using the reality of all numbers $\lambda_{n}$, we obtain

$\begin{array}[]{c}\int_{0}^{x}\langle f^{T}(s)F(s,t)ds,f^{T}(t)\rangle dt\\\
=\sum_{n=-\infty}^{\infty}\int_{0}^{x}\bigr{(}\int_{0}^{x}\\{z_{n}[f_{1}(s)\sin\lambda_{n}s\sin\lambda_{n}t-f_{2}(s)\cos\lambda_{n}s\sin\lambda_{n}t]\\\
-\frac{1}{\pi}[f_{1}(s)\sin ns\sin nt-f_{2}(s)\cos ns\sin
nt]\\}ds\bigr{)}\overline{f_{1}(t)}dt\\\
+\sum_{n=-\infty}^{\infty}\int_{0}^{x}\bigr{(}\int_{0}^{x}\\{z_{n}[-f_{1}(s)\sin\lambda_{n}s\cos\lambda_{n}t+f_{2}(s)\cos\lambda_{n}s\cos\lambda_{n}t]\\\
-\frac{1}{\pi}[-f_{1}(s)\sin ns\cos nt+f_{2}(s)\cos ns\cos
nt]\\}ds\bigr{)}\overline{f_{2}(t)}dt\\\
=\par\par\par\sum_{n=-\infty}^{\infty}\bigr{(}z_{n}\int_{0}^{x}[f_{1}(s)\sin\lambda_{n}s-f_{2}(s)\cos\lambda_{n}s]ds\int_{0}^{x}\sin\lambda_{n}t\overline{f_{1}(t)}dt\\\
-\frac{1}{\pi}\int_{0}^{x}[f_{1}(s)\sin ns-f_{2}(s)\cos ns]ds\int_{0}^{x}\sin
nt\overline{f_{1}(t)}dt\bigr{)}\\\
+\par\par\par\sum_{n=-\infty}^{\infty}\bigr{(}z_{n}\int_{0}^{x}[-f_{1}(s)\sin\lambda_{n}s+f_{2}(s)\cos\lambda_{n}s]ds\int_{0}^{x}\cos\lambda_{n}t\overline{f_{2}(t)}dt\\\
-\frac{1}{\pi}\int_{0}^{x}[-f_{1}(s)\sin ns+f_{2}(s)\cos ns]ds\int_{0}^{x}\cos
nt\overline{f_{2}(t)}dt\bigr{)}\\\
=\par\par\par\par\par\sum_{n=-\infty}^{\infty}z_{n}\bigr{(}\int_{0}^{x}[f_{1}(s)\sin\lambda_{n}s-f_{2}(s)\cos\lambda_{n}s]ds\int_{0}^{x}\sin\lambda_{n}t\overline{f_{1}(t)}dt\\\
+\int_{0}^{x}[-f_{1}(s)\sin\lambda_{n}s+f_{2}(s)\cos\lambda_{n}s]ds\int_{0}^{x}\cos\lambda_{n}t\overline{f_{2}(t)}dt\bigr{)}\\\
-\sum_{n=-\infty}^{\infty}\frac{1}{\pi}\bigr{(}\int_{0}^{x}[f_{1}(s)\sin ns-
f_{2}(s)\cos ns]ds\int_{0}^{x}\sin nt\overline{f_{1}(t)}dt\\\
+\int_{0}^{x}[-f_{1}(s)\sin ns+f_{2}(s)\cos ns]ds\int_{0}^{x}\cos
nt\overline{f_{2}(t)}dt\bigr{)}\\\
=\par\par\par\par\par\par\par\sum_{n=-\infty}^{\infty}z_{n}\bigr{(}\int_{0}^{x}[f_{1}(t)\sin\lambda_{n}t-f_{2}(t)\cos\lambda_{n}t]dt\int_{0}^{x}\sin\lambda_{n}t\overline{f_{1}(t)}dt\\\
+\int_{0}^{x}[-f_{1}(t)\sin\lambda_{n}t+f_{2}(t)\cos\lambda_{n}t]dt\int_{0}^{x}\cos\lambda_{n}t\overline{f_{2}(t)}dt\bigr{)}\\\
-\sum_{n=-\infty}^{\infty}\frac{1}{\pi}\bigr{(}\int_{0}^{x}[f_{1}(t)\sin nt-
f_{2}(t)\cos nt]dt\int_{0}^{x}\sin nt\overline{f_{1}(t)}dt\\\
+\int_{0}^{x}[-f_{1}(t)\sin nt+f_{2}(t)\cos nt]dt\int_{0}^{x}\cos
nt\overline{f_{2}(t)}dt\bigr{)}\\\
=\par\sum_{n=-\infty}^{\infty}z_{n}\int_{0}^{x}[f_{1}(t)\sin\lambda_{n}t-f_{2}(t)\cos\lambda_{n}t]dt\int_{0}^{x}[\overline{f_{1}(t)}\sin\lambda_{n}t-\overline{f_{2}(t)}\cos\lambda_{n}t]dt\\\
-\sum_{n=-\infty}^{\infty}\frac{1}{\pi}\int_{0}^{x}[f_{1}(t)\sin nt-
f_{2}(t)\cos nt]dt\int_{0}^{x}[\overline{f_{1}(t)}\sin
nt-\overline{f_{2}(t)}\cos nt]dt\\\
=\par\par\par\par\sum_{n=-\infty}^{\infty}z_{n}|\int_{0}^{x}\langle
f(t),Y_{0}(t,\lambda_{n})\rangle
dt|^{2}-\sum_{n=-\infty}^{\infty}\frac{1}{\pi}|\int_{0}^{x}\langle
f(t),Y_{0}(t,n)\rangle dt|^{2}.\par\par\par\end{array}$

It is well known that the function system
$\\{\frac{1}{\sqrt{\pi}}Y_{0}(t,n)\\}$ $(n\in\mathbb{Z})$ is an orthonormal
basis in $L_{2,2}(0,\pi)$, hence it follows from the Parseval equality that

$\|f\|_{L_{2,2}(0,x)}^{2}=\sum_{n=-\infty}^{\infty}\frac{1}{\pi}\left|\int_{0}^{x}\langle
f(t),Y_{0}(t,n)\rangle dt\right|^{2},$

consequently,

$\sum_{n=-\infty}^{\infty}z_{n}\left|\int_{0}^{x}\langle
f(t),Y_{0}(t,\lambda_{n})\rangle dt\right|^{2}=0.$

Since all the numbers $z_{n}$ are situated strictly in the same half-plane
relative to a line which passes through the origin, we see that

$\int_{0}^{x}\langle f(t),Y_{0}(t,\lambda_{n})\rangle dt=0$

for all $n\in\mathbb{Z}$. It follows from (3.12) that the function
$s(\lambda)$ is a sin-type function [10], therefore [1, Lemma 5.3], the system
$Y_{0}(t,\lambda_{n})$ is a Riesz basis of $L_{2,2}(0,\pi)$, hence the system
$Y_{0}(t,\lambda_{n})$ is complete in $L_{2,2}(0,\pi)$, it follows now that
$f(t)\equiv 0$.

By [26, Th. 5.1], the functions $c(\lambda)$ and $-s(\lambda)$ are the entries
of the first line of the monodromy matrix

$\tilde{E}(\pi,\lambda)=\begin{pmatrix}\tilde{c}_{1}(\pi,\lambda)&-\tilde{s}_{2}(\pi,\lambda)\\\
\tilde{s}_{1}(\pi,\lambda)&\tilde{c}_{2}(\pi,\lambda)\end{pmatrix}$

for problem (1.1), (2.1) with a potential $\tilde{V}\in L_{2}$, i.e.,

$c(\lambda)=\tilde{c}_{1}(\pi,\lambda),s(\lambda)=\tilde{s}_{2}(\pi,\lambda).$
$None$

The corresponding characteristic determinant

$\tilde{\Delta}(\lambda)=(-1)^{\theta+1}+(\tilde{c}_{1}(\pi,\lambda)+\tilde{c}_{2}(\pi,\lambda))/2=(-1)^{\theta+1}+\cos\pi\lambda+\tilde{f}(\lambda),$

where $\tilde{f}\in PW_{\pi}$. It follows from (2.3), (3.5), (3.6), (3.21)
that

$\begin{array}[]{c}\tilde{\Delta}(\lambda_{n})=(-1)^{\theta+1}+(\tilde{c}_{1}(\pi,\lambda_{n})+\tilde{c}_{2}(\pi,\lambda_{n}))/2\\\
=(-1)^{\theta+1}+(\tilde{c}_{1}(\pi,\lambda_{n})+\frac{1}{\tilde{c}_{1}(\pi,\lambda_{n})})/2=(-1)^{\theta+1}+(c(\lambda_{n})+\frac{1}{c(\lambda_{n})})/2\\\
=\par(-1)^{\theta+1}+\chi(\lambda_{n})=U(\lambda_{n}).\end{array}$

This implies that the function

$\Phi(\lambda)=\frac{U(\lambda)-\tilde{\Delta}(\lambda)}{s(\lambda)}=\frac{f(\lambda)-\tilde{f}(\lambda)}{s(\lambda)}$

is an entire function in the whole complex plane. Since by the Paley-Wiener
theorem

$|f(\lambda)-\tilde{f}(\lambda)|<C_{3}e^{\pi|{\rm Im}\lambda|},$ $None$

then by (3.13) $|\Phi(\lambda)|\leq C_{4}$ if $|{\rm Im}\lambda|\geq M$. We
denote by $H$ the set
$\Gamma(N_{0}+1/2,1/10)\bigcup\Gamma(-N_{0}-1/2,1/10))\bigcup\Gamma_{|n|>N_{0}}(n,1/10)$.
Since the function $s(\lambda)$ is a sin-type function [11], then
$|s(\lambda)|>C_{5}>0$ if $\lambda\notin H$. From this inequality, (3.22) and
the Maximum Principle we obtain that $|\Phi(\lambda)|<C_{6}$ in the strip
$|{\rm Im}\lambda|\leq M$, hence the function $\Phi(\lambda)$ is bounded in
the whole complex plane and, by virtue of Liouville theorem, it is a constant.
Let $|{\rm Im}\lambda|=M$, then it follows from (3.10) that
$\lim_{|\lambda|\to\infty}(f(\lambda)-\tilde{f}(\lambda))=0$, consequently
$\Phi(\lambda)\equiv 0$, therefore $U(\lambda)\equiv\tilde{\Delta}(\lambda)$.

Remark 3.1. Necessity of condition (3.1) for the Dirac operators with skew-
symmetric potentials by another method was established in [27].

3.2. Spectrum

Theorem 3.2. For a set $\Lambda$ to be the spectrum of some Dirac operator
(1.1), (2.1) with a complex-valued potential $V\in L_{2}(0,\pi)$ it is
necessary and sufficient that it consists of two sequences of eigenvalues
$\lambda_{n,j}$ satisfying condition (2.6) and the inequality

$\sum_{k=-\infty}^{\infty}\left|\sum_{n=-\infty}^{\infty}\frac{\varepsilon_{n,1}+\varepsilon_{n,2}}{2n-2k-1}\right|<\infty.$
$None$

Proof. The proof of the theorem is carried out in the same lines for periodic
and antiperiodic cases, and here we present the reasoning only for periodic
one.

1\. Sufficiency. Let two sequences $\lambda_{n,j}$ satisfy conditions (2.6)
and (3.23). Evidently, there exists a constant $M$ such that

$\sup_{n,j}|\varepsilon_{n,j}|<M,\quad\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\varepsilon_{n,j}|^{2}<M.$
$None$

It is well known that

$\sin\pi\lambda=\pi\lambda\prod_{n=-\infty\atop n\neq
0}^{\infty}\frac{n-\lambda}{n}=\pi\lambda\prod_{n=-\infty\atop n\neq
0}^{\infty}\left(1-\frac{\lambda}{n}\right),$

hence,

$\sin^{2}\frac{\pi\lambda}{2}=\frac{\pi^{2}\lambda^{2}}{4}\prod_{n=-\infty\atop
n\neq
0}^{\infty}\left(\frac{2n-\lambda}{2n}\right)^{2}=\frac{\pi^{2}\lambda^{2}}{4}\prod_{n=-\infty\atop
n\neq 0}^{\infty}\left(1-\frac{\lambda}{2n}\right)^{2},$

therefore the function $\Delta_{0}(\lambda)=-1+\cos\pi\lambda$ has the
representation

$\Delta_{0}(\lambda)=-\frac{\pi^{2}\lambda^{2}}{2}\prod_{n=-\infty\atop n\neq
0}^{\infty}\frac{(2n-\lambda)(2n-\lambda)}{4n^{2}}.$ $None$

Evidently,

$|\Delta_{0}(\lambda)|<c_{1}e^{\pi|{\rm Im}\lambda|}.$ $None$

Denote

$\Delta(\lambda)=-\frac{\pi^{2}}{2}(\lambda_{0,1}-\lambda)(\lambda_{0,2}-\lambda)\prod_{n=-\infty\atop
n\neq
0}^{\infty}\frac{(\lambda_{n,1}-\lambda)(\lambda_{n,2}-\lambda)}{4n^{2}}.$

Let $f(\lambda)=\Delta(\lambda)-\Delta_{0}(\lambda)$. Investigation of
properties of the function $f(\lambda)$ is based on the following
propositions.

Proposition 2.The function $f(\lambda)$ is an entire function of exponential
type not exceeding $\pi$.

Denote $\Gamma$ the union of the disks $\Gamma(2n,1/4)$ $(n\in Z)$. If
$\lambda\notin\Gamma$, then

$f(\lambda)=-\Delta_{0}(\lambda)\left(1-\frac{\Delta(\lambda)}{\Delta_{0}}\right)=-\Delta_{0}(\lambda)(1-\phi(\lambda)),$
$None$

where

$\begin{array}[]{c}\phi(\lambda)=\frac{(\lambda_{0,1}-\lambda)(\lambda_{0,2}-\lambda)}{\lambda^{2}}\sum_{n=-\infty\atop
n\neq
0}^{\infty}\frac{(\lambda_{n,1}-\lambda)(\lambda_{n,2}-\lambda)}{(2n-\lambda)(2n-\lambda)}\\\
=(1-\frac{\lambda_{0,1}}{\lambda})(1-\frac{\lambda_{0,2}}{\lambda})\prod_{n=-\infty\atop
n\neq
0}^{\infty}\prod_{j=1}^{2}(1+\frac{\varepsilon_{n,j}}{2n-\lambda})=\prod_{n=-\infty}^{\infty}\prod_{j=1}^{2}(1+\alpha_{n,j}(\lambda)),\end{array}$

where $\alpha_{0,j}(\lambda)=\frac{-\lambda_{0,j}}{\lambda}$,
$\alpha_{n,j}(\lambda)=\frac{\varepsilon_{n,j}}{2n-\lambda}$. Let us estimate
the function $\phi(\lambda)$. It follows from (3.24) that

$\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(\lambda)|\leq
c_{2}+\sum_{n=-\infty\atop n\neq
0}^{\infty}\sum_{j=1}^{2}(|\varepsilon_{n,j}|^{2}+|2n-\lambda|^{-2})/2<c_{3}.$
$None$

It is easy to see that for all $|n|>n_{0}$, where $n_{0}$ is a sufficiently
large number, we have

$|\alpha_{n,j}(\lambda)|<1/4$ $None$

for all $\lambda\notin\Gamma$. If $|n|\leq n_{0}$, then inequality (3.29)
holds for all sufficiently large $|\lambda|$, hence inequality (3.29) is valid
for all $|\lambda|\geq C_{0}$. It follows from (3.28), (3.29), and elementary
inequality

$|\ln(1+z)|\leq 2|z|,$ $None$

which is valid if $|z|\leq 1/4$ that

$\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\ln(1+\alpha_{n,j}(\lambda))|\leq
c_{4}.$

Here and throughout the following, we choose the branch of $\ln(1+z)$ that is
zero for $z=0$. In view of [9, p. 433], we rewrite the last relation in the
form

$|\phi(\lambda)|\leq\prod_{n=-\infty}^{\infty}\prod_{j=1}^{2}|1+\alpha_{n,j}(\lambda)|\leq
e^{c_{4}}.$ $None$

It follows from (3.26), (3.27), (3.31) that

$|f(\lambda)|<c_{5}e^{\pi|{\rm Im}\lambda|}$ $None$

outside the domain $\Gamma^{\prime}=\Gamma\cup\\{|\lambda|<C_{0}\\}$. In
particular, inequality (3.32) is valid if $\lambda$ belongs lines ${\rm
Im}\lambda=\pm C_{0}$ and vertical segments with vertexes
$(2k-1,-C_{0}),(2k-1,C_{0})$, $|2k-1|>C_{0}$, $k\in\mathbb{Z}$. By the Maximum
Principle, inequality (3.32) holds in whole complex plane, hence the function
$f(\lambda)$ is an entire function of exponential type not exceeding $\pi$.

Proposition 3.The function $f$ belongs to $PW_{\pi}$.

Denote

$W(\lambda)=\ln\phi(\lambda)=\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}\ln(1+\alpha_{n,j}(\lambda)),$

then

$f(\lambda)=-\Delta_{0}(\lambda)\left(1-e^{W(\lambda)}\right).$ $None$

Let us estimate the function $W(\lambda)$ if $\lambda\notin\Gamma^{\prime}$.
It follows from (3.24), (3.29), (3.30) that

$\begin{array}[]{c}|W(\lambda)|\leq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\ln(1+\alpha_{n,j}(\lambda))|\leq
2\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(\lambda)|\\\
\leq\frac{2M}{|\lambda)|}+\sum_{n=-\infty\atop n\neq
0}^{\infty}\sum_{j=1}^{2}(\frac{|\varepsilon_{n,j}|^{2}}{10M}+\frac{10M}{|2n-\lambda|^{2}})\leq\frac{2M}{|\lambda|}+1/10+20M\sum_{n=0}^{\infty}\frac{1}{n^{2}+|{\rm
Im}\lambda|^{2}}\\\ \leq\frac{2M}{|\lambda|}+1/10+20M(\frac{2}{|{\rm
Im}\lambda|^{2}}+\int_{1}^{\infty}\frac{dx}{x^{2}+|{\rm
Im}\lambda|^{2}})\leq\frac{2M}{|{\rm Im}\lambda|}+1/10+20M(\frac{2}{|{\rm
Im}\lambda|^{2}}+\frac{\pi}{2|{\rm Im}\lambda|}).\end{array}$

The last inequality implies that

$|W(\lambda)|<1/4$ $None$

if $|{\rm Im}\lambda|\geq M_{1}=10(\pi+2+22M)+C_{0}$. Then from the trivial
inequality

$\frac{|z|}{2}\leq|1-e^{z}|\leq 2|z|,$ $None$

which holds for $|z|\leq 1/4$, we obtain the inequality
$|1-e^{W(\lambda)}|\leq 2|W(\lambda)|$, which, together with (3.26) and (3.33)
implies that

$|f(\lambda)|\leq c_{6}|W(\lambda)|$ $None$

for $\lambda\in l$, where $l$ is the line ${\rm Im}\lambda=M_{1}$. Let us
prove that

$\int_{l}|W(\lambda)|^{2}d\lambda<\infty.$ $None$

From the elementary inequality $|\ln(1+z)-z|\leq|z|^{2}$ true for $|z|\leq
1/2$, we obtain

$\ln(1+z)-z=r(z),$

where $|r(z)|\leq|z|^{2}$, hence,

$W(\lambda)=S_{1}(\lambda)+S_{2}(\lambda),$ $None$

where

$S_{1}(\lambda)=\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}\alpha_{n,j}(\lambda),$

$|S_{2}(\lambda)|\leq\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(\lambda)|^{2}.$

Evidently,

$|W(\lambda)|\leq|S_{1}(\lambda)|+|S_{2}(\lambda)|.$ $None$

Set

$I_{m}=\int_{l}|S_{m}(\lambda)|^{2}d\lambda$

$(m=1,2)$. First consider the integral $I_{1}$. It follows from [25] that

$I_{1}=\int_{l}\left|\sum_{n=-\infty}^{\infty}\frac{\varepsilon_{n,1}+\varepsilon_{n,2}}{2n-\lambda}\right|^{2}d\lambda<\infty.$
$None$

It is readily seen that

$|S_{2}(\lambda)|\leq\sum_{n=-\infty}^{\infty}\frac{|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2}}{|2n-\lambda|^{2}}<c_{7},$

hence,

$\begin{array}[]{c}I_{2}\leq
c_{7}\int_{l}\left(\sum_{n=-\infty}^{\infty}\frac{|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2}}{|2n-\lambda|^{2}}\right)d\lambda=c_{8}\sum_{n=-\infty}^{\infty}(|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2})\int_{l}\frac{d\lambda}{|2n-\lambda|^{2}}\\\
<c_{9}\sum_{n=-\infty}^{\infty}(|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2})<c_{10}.\end{array}$
$None$

Relations (3.39-3.45) imply (3.37). It follows from (3.36), (3.37), and [22,
p. 115] that

$\int_{R}|f(\lambda)|^{2}d\lambda<\infty.$ $None$

Proposition 4.The function $f(\lambda)$ satisfies condition (3.1).

Obviously, $\Delta_{0}(2k)=0$, hence $f(2k)=\Delta(2k)$. Since the function
$\Delta(\lambda)$ is bounded in the strip $|{\rm Im}\lambda|\leq 1$, and for
all sufficiently large $|k|$ the inequality $|\varepsilon_{k,j}|<1/2$ takes
place, by the Maximum Principle, we have

$|f(2k)|=|\Delta(2k)|\leq|\varepsilon_{k,1}||\varepsilon_{k,2}|\max_{|2k-\lambda|=1}\left|\frac{\Delta(\lambda)}{(\lambda_{k,1}-\lambda)(\lambda_{k,2}-\lambda)}\right|\leq
c_{11}(|\varepsilon_{k,1}|^{2}+|\varepsilon_{k,2}|^{2}).$ $None$

Let us estimate $|f(2k+1)|$. Obviously, $\Delta_{0}(2k+1)=-2$. Denote

$\epsilon_{n}=\max(|\varepsilon_{n,1}|,|\varepsilon_{n,2}|).$

There exists a number $n_{0}>0$ such that

$\sum_{|n|>n_{0}}\epsilon_{n}^{2}<1/1000,$

and for any $|n|>n_{0}$ the inequality $\epsilon_{n}^{2/3}<1/1000$ holds. Let
$\lambda\notin\Gamma^{\prime}$. Supplementary suppose that

$|\lambda|>M_{2}=1000(2n_{0}+1)n_{0}M.$

Then, using the well known inequality $ab\leq\frac{a^{p}}{p}+\frac{b^{q}}{q}$
$(a,b>0,p,q>1,1/p+1/q=1)$, we obtain

$\begin{array}[]{c}\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(\lambda)|\leq
2(\sum_{|n|\leq
n_{0}}\frac{\epsilon_{n}}{|2n-\lambda|}+\sum_{|n|>n_{0}}\frac{\epsilon_{n}}{|2n-\lambda|})\\\
\leq 2M\sum_{|n|\leq
n_{0}}\frac{1}{|2n-\lambda|}+2\sum_{|n|>n_{0}}(\epsilon_{n}^{2}+\frac{\epsilon_{n}^{2/3}}{|2n-\lambda|^{4/3}})\\\
\leq 1/50+1/500\sum_{n=1}^{\infty}\frac{1}{n^{4/3}}<1/10,\end{array}$ $None$

hence inequality (3.34) is valid for all $\lambda$ belonging to the considered
domain. Arguing as above, we see that

$|f(\lambda)|\leq
c_{12}\left(|\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}\alpha_{n,j}(\lambda)|+\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(\lambda)|^{2}\right).$

The last inequality implies that for all $|2k+1|>k_{0}$, where
$k_{0}=\max(C_{0},M_{2})$,

$|f(2k+1)|\leq
c_{13}\left(\left|\sum_{n=-\infty}^{\infty}\frac{\varepsilon_{n,1}+\varepsilon_{n,2}}{2n-2k-1}\right|+\sum_{n=-\infty}^{\infty}\frac{|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2}}{|2n-2k-1|^{2}}\right).$
$None$

Clearly,

$\begin{array}[]{c}\sum_{k=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\frac{|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2}}{|2n-2k-1|^{2}}=\sum_{n=-\infty}^{\infty}(|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2})\sum_{k=-\infty}^{\infty}\frac{1}{|2n-2k-1|^{2}}\\\
<c_{14}\sum_{n=-\infty}^{\infty}(|\varepsilon_{n,1}|^{2}+|\varepsilon_{n,2}|^{2})<c_{15}.\end{array}$
$None$

It follows from (3.23),(3.43), (3.45), (3.46) that (3.1) holds, hence the
function $f(\lambda)$ satisfies all conditions of Theorem 3.1, and the
function $\Delta(\lambda)$ is the characteristic determinant of some problem
(1.1), (2.1).

2\. Necessity. If a set $\\{\Lambda\\}$ is the spectrum of a Dirac operator
(1.1), (2.1), then relation (2.6) takes place [4, Th. 6.5]. Let us prove that
condition (3.23) holds. Since
$f(\lambda)=\Delta(\lambda)-\Delta_{0}(\lambda)$, then by Theorem 3.1 relation
(3.1) is valid.

Let $\lambda=2k+1$, $k\in\mathbb{Z}$, $|2k+1|>k_{0}$, hence inequality (3.44)
holds. Since $\Delta_{0}(2k+1)=-2$, it follows from (3.33) and (3.35) that

$|W(2k+1)|\leq|f(2k+1)|.$

This, together with (3.38) implies

$|S_{1}(2k+1)|\leq|f(2k+1)|+\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(2k+1)|^{2}.$
$None$

Using (3.46), we find that

$\sum_{n=-\infty}^{\infty}\sum_{j=1}^{2}|\alpha_{n,j}(2k+1)|^{2}<c_{16}.$
$None$

It follows from (3.47), (3.48), and (3.1) that

$\sum_{|2k+1|>k_{0}}|S_{1}(2k+1)|<c_{17}.$

It is easy to see that

$\sum_{|2k+1|\leq k_{0}}|S_{1}(2k+1)|<k_{0}c_{18}.$

The last two inequalities imply (3.23).

3\. Example. We give an example when (2.6) holds but (3.23) does not hold. Let
$\varepsilon_{n,1}=\varepsilon_{n,2}=1/m$ if $n=2^{m}$ and
$\varepsilon_{n,1}=\varepsilon_{n,2}=0$ if $n\neq 2^{m}$, $m=1,2,\ldots$,
$n\in\mathbb{Z}$. Denote

$\gamma_{k}=\sum_{n=-\infty}^{\infty}\frac{\varepsilon_{n,1}+\varepsilon_{n,2}}{2n-2k-1},\quad
k\in\mathbb{Z},$

hence,

$\gamma_{k}=2\sum_{m=1}^{\infty}\frac{1}{m(2^{m+1}-2k-1)}.$

Let $k=2^{p}$, $p\in\mathbb{N}$, then

$\gamma_{2^{p}}=2\sum_{m=1}^{\infty}\frac{1}{m(2^{m+1}-2^{p+1}-1)}=-\frac{2}{p}+\sigma_{p,1}+\sigma_{p,2},$

where

$\sigma_{p,1}=2\sum_{m=1}^{p-1}\frac{1}{m(2^{m+1}-2^{p+1}-1)},\quad\sigma_{p,2}=2\sum_{m=p+1}^{\infty}\frac{1}{m(2^{m+1}-2^{p+1}-1)}.$

A simple computation shows that

$|\sigma_{p,1}|\leq\frac{1}{2^{p}}\sum_{m=1}^{p-1}\frac{1}{m(1-2^{m-p}-1/2^{p+1})}\leq\frac{4}{2^{p}}\sum_{m=1}^{p-1}\frac{1}{m}\leq\frac{4(1+\ln
p)}{2^{p}}$ $None$

and

$|\sigma_{p,2}|\leq
2\sum_{l=1}^{\infty}\frac{1}{(l+p)(2^{l+p+1}-2^{p+1}-1)}\leq\frac{1}{p2^{p}}\sum_{l=1}^{\infty}\frac{1}{2^{l}-1-1/2^{p+1}}\leq\frac{4}{p2^{p}}.$
$None$

It follows from (3.49), (3.50) that $|\sigma_{p,1}+\sigma_{p,2}|\leq 1/p$ if
$p\geq 10$, hence $|\gamma_{2^{p}}|>1/p$, therefore the series

$\sum_{k\in\mathbb{Z}}|\gamma_{k}|$

diverges.

References

[1] A. Albeverio, R. Hryniv, and Ya. Mykytyuk, Inverse spectral problems for
Dirac operators with summable potentials, Russ. J. Math. Phys. 12 (2005), no.
4, 406-423.

[2] N. Bondarenko, S. Buterin An inverse spectral problem for integro-
differential Dirac operators with general convolution kernels, Appl. Analysis
99 (2020), no. 4, 700-716.

[3] Vasil B. Daskalov and Evgeni Kh. Khristov Explicit formulae for the
inverse problem for the regular Dirac operator, Inverse Problems 16 (2000),
247258.

[4] P. Djakov and B. Mityagin, Unconditional Convergence of Spectral
Decompositions of 1D Dirac operators with Regular Boundary Conditions, Indiana
Univ. Math. J. 61 (2012), 359-398.

[5] T.T. Dzabiev, The inverse problem for the Dirac equation with a
singularity, Acad. Nauk. Azerbaidzan. SSR. Dokl. 22, (1966), no. 11, p. 8-12.
(in Russian).

[6] M.G. Gasymov, T.T. Dzabiev, Solution of the inverse problem by two spectra
for the Dirac equation on a finite interval, Acad. Nauk. Azerbaidzan. SSR.
Dokl. 22, (1966), no. 7, p. 3-7. (in Russian).

[7] F. Gesztesy, A. Sakhnovich, The inverse approach to Dirac-type systems
based on the A-function concept, Journal of Functional Analysis, Volume 279,
Issue 6, 1 October 2020, 108609.

[8] O. Gorbunov O., V. Yurko, Inverse problem for Dirac system with
singularities in interior points, Anal Math. Phys. bf 6, (2016), no. 1, 1-29.

[9] M.A. Lavrentiev, B.V. Shabat Methods of Theory of Complex Variable, Nauka,
Moscow, 1973 (in Russian).

[10] B. Ya. Levin, Lectures on Entire Functions, Amer. Math. Soc. Transl.
Math. Monographs 150, Amer. Math. Providence, RI, 1996.

[11] B. Ya. Levin, I. V. Ostrovskii, On small perturbations of the set of
zeros of functions of sine type, Math. USSR-Izv. 14, (1980), no. 1, 79101.

[12] B.M. Levitan, I.S. Sargsyan, SturmLiouville and Dirac operators, Kluwer
Academic Publishers, Dordrecht, 1991.

[13] M. M. Malamud, On Borg-type theorems for first-order systems on a finite
interval, Funktsional. Anal. i Prilozhen. 33 (1999), no. 1, 75-80 (Russian);
Engl. transl.: Funct. Anal. Appl. 33 (1999), no. 1, 64-68.

[14] M. M. Malamud, Uniqueness questions in inverse problems for systems of
differential equations on a finite interval, Trans. Moscow Math. Soc. 60
(1999), 173224.

[15] S.G. Mamedov, The inverse boundary problem on a finite interval for the
Dirac’s systems of equations, Azerbaidzan Gos. Univ. Uchen. Zap. Ser. Fiz-Mat.
Nauk. (1975), no. 5, 61-67 (in Russian).

[16] V. A. Marchenko, SturmLiouville operators and their applications,
Birkhauser Verlag, Basel, 1986.

[17] T.V. Misyura, Characterization of spectra of periodic and anti-periodic
problems generated by Dirac’s operators. II, Theoriya functfii, funct. analiz
i ikh prilozhen. 31 (1979), 102-109.

[18] Y. V. Mykytyuk, D.V. Puyda, Inverse spectral problems for Dirac operators
on a finite interval, J. Math. Anal. Appl. 386 (2012), no. 1, 177194.

[19] I.M. Nabiev, Solution of the class of inverse problems for the Dirac
operators, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. Mech. 21
(2001), 146-157.

[20] I.M. Nabiev,Characteristic of spectral data of Dirac operators, Trans.
Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. Mech. 24 (2004), 161-166.

[21] I.M. Nabiev, Solution of the quasiperiodic problem for the Dirac system,
Math. Notes, 89 (2011), no. 6, 845-852.

[22] I.M. Nabiev, The inverse periodic problem for the Dirac operator,
Proceeding of IMM of NAS of Azerbaijan, XIX (2003), 177-180.

[23] S.M. Nikolskii, Approximation of Functions of Several Variables and
Embedding Theorems, Nauka, Moscow, 1977 (in Russian).

[24] W. Ning, An inverse spectral problem for a nonsymmetric differential
operator: Reconstruction of eigenvalue problem, J. Math. Anal. Appl. 327
(2007), no. 2, 1396-1419.

[25] J.-J. Sansug, V. Tkachenko, Characterization of the periodic and
antiperiodic spectra of nonselfadjoint Hill’s operators, Oper. Theory Adv. and
Appl. 98 (1997), 216-224.

[26] V. Tkachenko, Non-self-adjoint Periodic Dirac Operators, Oper. Theory:
Adv. and Appl. 123 (2001), 485-512.

[27] V. Tkachenko, Non-self-adjoint Periodic Dirac Operators with finite-band
spectra, Int. Equ. Oper. Theory 36 (2000), 325-348.

[28] C.-Fu Yang and V. Yurko, Recovering Dirac operator with nonlocal boundary
conditions, J. Math. Anal. Appl. 440 (2016), no 1, 156-166.

[29] V.A. Yurko, Method of spectral mappings in the inverse problem theory,
Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002.

[30] V.A. Yurko, Introduction to the Theory of Inverse Spectral Problems,
Fizmatlit, Moscow, 2007 (in Russian).

[31] V.A. Yurko Inverse spectral problems for differential operators and their
applications, Gordon and Breach Science Publishers, Amsterdam, 2000.

[32] V.A.Yurko, An inverse spectral problem for singular non-selfadjoint
differential systems, Sbornik: Mathematics 195 (2004), no. 12, 18231854.

[33] V.A. Yurko Inverse spectral problems for differential systems on a finite
interval, Results Math. 48 (2005), no. (34), 371386.

[34] V.A. Yurko An inverse problem for differential systems on a finite
interval in the case of multiple roots of the characteristic polynomial, Diff.
Eqns. 41 (2005), no. 6, 818823.

[35] V.A. Yurko, An inverse problem for differential systems with multiplied
roots of the characteristic polynomial, J. Inv. Ill-Posed Probl. 13 (2005),
no. 5, 503512.

Russian Technological University, Prospect Vernadskogo 78, Moscow, 119454,
Russia

email<EMAIL_ADDRESS>